Quantum Tunnelling of a Molecule with a Single Bound State Jeremy J. Kavka B.Sc, University of Northern British Columbia, 2007 Thesis Submitted in Partial Fulfillment of The Requirements for the Degree of Master of Science in Mathematical, Computer, and Physical Sciences (Physics) University of Northern British Columbia May 2009 ©Jeremy J. Kavka, 2009 1*1 Library and Archives Canada Bibliotheque et Archives Canada Published Heritage Branch Direction du Patrimoine de I'edition 395 Wellington Street OttawaONK1A0N4 Canada 395, rue Wellington OttawaONK1A0N4 Canada Your file Votre reference ISBN: 978-0-494-60825-8 Our file Notre reference ISBN: 978-0-494-60825-8 NOTICE: AVIS: The author has granted a nonexclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or noncommercial purposes, in microform, paper, electronic and/or any other formats. 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Conformement a la loi canadienne sur la protection de la vie privee, quelques formuiaires secondaires ont ete enleves de cette these. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. Bien que ces formuiaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. 1+1 Canada Abstract We investigate, in one spatial dimension, the quantum mechanical tunnelling of a diatomic, homogeneous molecule with a single bound state incident upon an external barrier. Both time-independent and time-dependent tunnelling are investigated, using analytical and numerical methods. In the time-independent case, we first derive a formal solution for the molecule's wave function. Then, using the method of variable reflection and transmission amplitudes, we find that the probabilities of reflection and transmission in the bound state decrease with decreased binding strength, while the probabilities of refection and transmission in an unbound state increase with decreased binding energy. In the time-dependent case, we consider a molecule with discrete unbound states. The molecular wave function is modelled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration. It is found that, in addition to reflecting and transmitting, the molecule may also straddle the potential barrier in an unbound state. ii Contents Abstract ii Contents iii List of Figures vii Acknowledgement ix 1 Introduction 1 1.1 History of Studies in Molecular Tunnelling 1 1.2 The Case of the Molecule with a Single Bound State 5 1.3 Outline of Thesis 7 2 3 Formulation of t h e Problem 11 2.1 The Time-independent Case 12 2.2 The Time-dependent Case 18 2.3 Summary 21 Formal Solution t o the Time-independent Multichannel Schrodinger iii 4 Equation 22 3.1 The Lippmann-Schwinger Equation 23 3.2 The Formal Solution to the Multichannel Schrodinger Equation 25 3.3 Calculation of Probabilities 31 3.4 Summary 35 Preliminary Analytical Results 36 4.1 37 A Special Case: The Delta Well Binding Potential 4.1.1 Delta Well Potential with Finite Interaction Range 4.1.2 39 Delta Well Potential with Infinite Interaction Range 42 4.2 Molecule Incident Upon an Infinite Barrier 4.3 Probability of Reflection and Transmission in the Bound State 44 for a Molecule Incident in the Bound State Upon a Delta Potential Barrier 4.3.1 4.3.2 4.4 5 54 Effective Potential for a Molecule of Definite Parity Incident Upon a Delta Barrier 55 Reflection and Transmission Amplitudes 57 Summary 60 The M e t h o d of Variable Reflection and Transmission Amplitudes 62 5.1 63 Discrete Unbound States iv 6 5.1.1 Lemma I 66 5.1.2 Lemma II 68 5.1.3 Lemma III 69 5.1.4 Final Result 69 5.2 Continuous Unbound States 72 5.3 Summary 76 Numerical Results for the Time-independent Case 78 6.1 The Double Well Potential 79 6.1.1 Relative Motion Eigenstates and Energies 80 6.1.2 The External Barrier; Dimensionless Quantities . . . . 83 6.1.3 Effective Potentials for a Molecule Incident Upon a 6.1.4 Delta Barrier 84 The Critical Wave Number 87 6.2 Numerical Method 89 6.3 Results of the Numerical Calculation 90 6.3.1 Probability of Reflection in the Bound State 91 6.3.2 Probability of Transmission in the Bound State 6.3.3 Probability of Reflection in an Unbound State 6.3.4 Probability of Transmission in an Unbound State 6.3.5 Total Probability of Transmission 6.3.6 Comparison of Numerical Results to the Analytical Predictions .... 93 95 ... 96 97 99 v 6.4 7 Summary Time-dependent Tunnelling 101 7.1 The Double Well Potential with Finite Interaction Range . . . 102 7.2 Numerical Method 106 7.3 Results of the Numerical Analysis 108 7.3.1 7.4 8 100 Reflection in the Bound State and in an Unbound State vs. k Ill 7.3.2 Analysis of Molecular Straddling 114 7.3.3 Probabilities as Functions of k 118 Summary 121 Conclusion 122 8.1 Summary of this Work 122 8.2 Future Work 127 A Normalization Constants for Relative Motion Eigenstates 129 A.l Normalization Constants for a Molecule with Discrete Unbound States 129 A.2 Normalization Constants for a Molecule with Continuous Unbound States 131 Bibliography 132 VI List of Figures 4.1 Plots of/^ s (a(a;)) and/ r ./ ls (a(a;)) for various values of x. . . . 49 6.1 Plot of the double-well binding potential 80 6.2 Plot of the effective potentials ZQQ(X) for two binding energies. 84 6.3 Plot of the effective potentials Z0q(x) 85 6.4 Plot of P^h vs. k for two binding energies 91 6.5 Plot of Pj.b vs. k for two binding energies 94 6.6 Plot of PRu vs. k for two binding energies 95 6.7 Plot of P®u vs. k for two binding energies 97 6.8 Plot of P?> vs. k for three binding energies 98 7.1 Plot of ZQO(x), Z0i(x), 7.2 Plots of PRb(l) for all A; 7.3 Plots of (1 - PRb(t)), 7.4 and Zu(x) 105 112 PTb(t), PRu{i), and PTu(i) for k = 1.00. . 113 Plots of {Pb^u(i) - PSu{i)) for all k 115 7.5 Plots of PSu{i), Pi^u(i), 7.6 Plots of {PsbiJ)) for all (1 - PRb{t)) and Prh{i) for all k. . . . 116 fc vii 118 7.7 (1 - PRh{i)), PTh(t), PRu(t), PTu(t) and PSu{l) at t = 200 with respect to k 119 vin Acknowledgement My deepest thanks must go, first and foremost, to my supervising professor Mark R.A. Shegelski. Without Professor Shegelski's encouragement, advice, and support, this thesis would not have been possible. It has been a privilege to work with Professor Shegelski during my masters studies. I would also like to extend thanks to the members of my supervisory committee, Professor Matthew Reid and Professor Lee Keener. Thanks must also be given to Professor Reid and Mr. Jeff Hnybida for their advice concerning numerical analysis. I further acknowledge Mr. Hnybida for having derived the extension of Razavy's method to the continuum, and for creating the computer program used to numerically integrate the resulting equations. I thank Dr. Y. Q. Wang for assistance with use of the High Performance Computing facility at the University of Northern British Columbia. I thank the Natural Sciences and Engineering Research Council of Canada for support in the form of an Alexander Graham Bell Scholarship and to the University of Northern British Columbia for numerous scholarships, awards, and teaching assistantships. Finally, I would like to thank my parents, Darlene and Frantisek Kavka, and my sister, Naomi Kavka, for their support throughout my studies. IX Chapter 1 Introduction 1.1 History of Studies in Molecular Tunnelling Quantum tunnelling is a phenomenon in which a microscopic object (e.g. an atom, molecule, or subatomic particle) passes through a potential barrier of greater magnitude than the total energy of the tunnelling object. According to the laws of classical physics, such tunnelling is not possible, i.e. it is, as the name suggests, a uniquely quantum mechanical phenomenon. The study of quantum tunnelling, indeed, began almost immediately following the formulation of quantum mechanics in the 1920's with Gamow's attempt, in 1928, to explain a-decay. Since then, quantum tunnelling has been employed to describe cold emission of electrons from a metallic surface and the electron current flow in semiconductors. Razavy provides a brief history of these important results in reference [1]. 1 Studies of quantum tunnelling have typically focussed on the case of single particles travelling in one dimension. The extension to the case of tunnelling in multiple dimensions, or to the case of the tunnelling of objects with internal degrees of freedom, is far from straightforward. The study of the quantum tunnelling of molecules, which is studied in this thesis, is a relatively new area of research. The first examination of molecular tunnelling did not occur until 1994, with Saito and Kayanuma's investigation of the resonant tunnelling of a pair of bound particles upon a single external barrier [2]. In their work, Saito and Kayanuma modelled the tunnelling molecule as a pair of point particles bound by a square well with a hard core. Using this model of the molecule, they calculated the probability of transmission of the molecule when incident upon a rectangular barrier. Saito and Kayanuma observed the occurrence of resonances in the calculated transmission spectrum, which they interpreted as the result of quasi-bound states of the centre of mass motion around the potential barrier. They also observed inelastic tunnelling due to transitions of the relative motion state caused by the tunnelling of the centre of mass through the potential barrier. Saito and Kayanuma's work was continued by Pen'kov [3, 4] in 2000. Pen'kov used a pair of point particles in one dimension, bound by a harmonic oscillator potential, as a model for a diatomic molecule. Like Saito and Kayanuma, Pen'kov observed resonances in the probability of transmission, with probabilities approaching unity for certain energies. Pen'kov explained this phenomenon as a result of interference suppression of the reflected wave, 2 similar to phenomena observed in optical behaviour. The tunnelling of a diatomic molecule in one [5] and three [6] dimensions was reported by Goodvin and Shegelski in 2005. Their work modelled the molecular binding potential as a double square well potential with a central barrier. They chose this binding potential because it was more effective at capturing the physical behaviour of molecules than the single well models employed in previous work. Theirs was also the first investigation to consider multiple relative motion states, and hence the effect of molecular transitions on tunnelling. In addition, their work was the first in the study of molecular tunnelling to employ Razavy's method of variable reflection and transmission amplitudes, which is described in reference [1]. This method allowed for a great simplification in the computational work needed to obtain reflection and transmission probabilities. Bulatov and Kornilovitch [7], also in 2005, developed a general, nonperturbative method to solve scattering problems for bound pairs of particles on a lattice. Applying this method to the case of a particle pair tunnelling through a weak link on a one dimensional lattice, they observed that tunnelling probabilities for certain momentum values were far higher than those for single particles at the same momentum values. Their result was consistent with the earlier findings of resonances in transmission probability. In 2006, Bacca and Feldmeir [8], and separately Lee [9], studied the resonant tunnelling of a pair of particles using the same formalism as Goodvin and Shegelski. Their work, however, made use of a binding potential consisting 3 of a single square well, rather than a double square well. This form of the binding potential is more applicable to nuclear physics than it is to molecular physics. Bertulani, Flambaum, and Zelevinsky [10], in 2007, showed that the probability of tunnelling of a pair of bound particles is significantly affected by the intrinsic structure of the particle pair. Specifically, they found that a particle, initially in spin state "up," subjected to a weak magnetic field inducing in the particle a spin "down" component, will have a significantly greater probability of tunnelling past a barrier than a particle not subjected to such a magnetic field. In 2008, Shegelski, Hnybida, Friesen, Lind, and Kavka [11] studied the tunnelling of a molecule with a continuum of unbound states. This was significant in that it was the first instance of the examination of molecular break-up upon a potential barrier. It was also the first study that considered a molecule with a continuum of relative motion states (previous studies had considered only discrete relative motion states). Also in 2008, Hnybida and Shegelski [12] studied the tunnelling of a molecule with many bound states. They found that the transmission probability was drastically altered by both the number of bound states and by the separation between bound state energies. Later in 2008, Shegelski, Hnybida, and Vogt [13] studied the formation of a molecule incident upon a potential barrier. That is, they calculated the probability of two unbound particles combining into a single bound molecule upon contact with a potential barrier. They found that the probability of molecule formation depended heavily on the strength of the external barrier. 4 All three of these papers made use of Razavy's method of variable reflection and transmission amplitudes. Recent technological advances have allowed for the experimental study of tunnelling of objects with internal degrees of freedom. In 2000, Lauhon and Ho directly observed the tunnelling of a single hydrogen atom on a metal surface by means of a scanning tunnelling microscope [14]. Direct observation of tunnelling in a single bosonic Josephson junction was reported by Albiez, Gati, Foiling, Hunsmann, Cristiani, and Oberthaler in 2005 [15]. Albiez et al. found that the tunnelling rate was drastically increased due to interactions between the tunnelling atoms. Previous work in this area, by contrast, had focussed on non-interacting atoms. The tunnelling of Cooper pairs was observed by Toppari, Kuhn, Halvari, Kinnunen, Leskinen, and Paraoanu in 2007 [16]. Resonant tunnelling of the Cooper pairs was observed, as predicted by the theoretical work mentioned earlier. 1.2 The Case of the Molecule with a Single Bound State Part of the work discussed in this thesis considers the case of a molecule with a single, increasingly weakly bound state, tunnelling past an external potential barrier in one dimension. This work differs from previous work in that it considers only a single bound state with access to unbound states. Previous work considered either the tunnelling of molecules with multiple 5 bound states or, in the earlier work, tunnelling of molecules with only a single state. The case of the tunnelling of a weakly bound molecule is an interesting one in that it presents an example of a seemingly "easy" problem which, upon further investigation, turns out to be quite difficult. Specifically, it was expected that, in the limit of an arbitrarily weak binding potential, the equations for the reflection and transmission coefficients would assume an analytically simple form, or would at least give clear values for the probabilities. This has turned out not to be the case. An example of a result that is different than what would have been expected is presented in chapter 4. We consider, as a simple test case, a molecule with a simple, arbitrarily weak binding potential incident upon an infinite barrier. It is clear that a molecule incident upon such a barrier must be reflected from the barrier, i.e. the probability of reflection is unity, and the probability of transmission is zero. Moreover, for an arbitrarily weakly bound molecule, we would expect the probability of molecular break-up upon reflection to approach unity as well, i.e. the probability of reflection in a bound state approaches zero as the molecule becomes more and more weakly bound. It was our initial hope to demonstrate this rigorously, in part to determine whether analysis really does become simplified in the case of weak molecular binding, but also in the hope that our analysis of this case could yield results useful for the analysis of a molecule incident upon a finite barrier. We were able to demonstrate, by analytical means, that the bound state of 6 the molecule must cease to exist once the molecule comes within a certain critical distance of the barrier. However, we also found that the equations describing the molecular wave function for this "simple" case are actually more complicated than the corresponding equations for a molecule incident upon a finite barrier. These difficulties served to illustrate the surprisingly interesting nature of the weakly bound molecule. Work on the problem of a weakly bound molecule incident upon an infinite barrier is ongoing. For the purposes of this thesis, we present results on the tunnelling of a molecule with a single bound state. We present time-independent and time-dependent formulations of the problem in chapter 2. Chapters 3 through 6 are concerned with the tunnelling behaviour of a molecule with respect to the binding strength of the molecule, while chapter 7 is concerned with the time-dependent tunnelling of a molecule. 1.3 Outline of Thesis In chapter 2, we present the formulation of the problem of a molecule incident upon an external potential barrier. The Hamiltonian operator is expressed in terms of the relative and centre-of-mass (CM) coordinates. The relative motion is shown to be dependent only on the choice of molecular binding potential, while the CM motion is found to be dependent on both the relative motion and the external potential barrier. We derive a set of equations known 7 collectively as the multi-channel Schrodinger equations, whose solution gives the wave function of a molecule incident upon a potential barrier. We derive multi-channel Schrodinger equations for both the time-independent case and the time-dependent case. The time-independent formulation is considered in chapters 3-6, and the time-dependent formulation is considered in chapter 7. In chapter 3, we derive a formal solution to the time-independent multichannel Schrodinger equation using Green's function methods. From the formal solution we obtain expressions for the reflection and transmission coefficients, as well as the probabilities of reflection and transmission. In chapter 4, we present analytical results obtained for the special case of a molecule with a delta well potential, a mathematically simple potential well. We consider the case of the molecule incident upon an infinite barrier. We described this work earlier in the introduction. As well, we present two additional results in this chapter. The first result, part of an analysis that as of this writing is still in progress, is an approximate, asymptotic solution to the molecular wave function incident upon an infinite barrier. The second is an argument showing that if a molecule, incident upon a potential barrier in the bound state, is arbitrarily weakly bound, the probability of reflection in the bound state approaches zero, while the probability of transmission in the bound state approaches unity. In chapter 5, we use a powerful technique developed by Razavy [1], known as the method of variable reflection and transmission amplitudes, to obtain a set of coupled differential equations describing the reflection and transmis8 sion coefficients defined in chapter 3. This allows us to numerically calculate the reflection and transmission coefficients without having to solve for the associated wave function, thus considerably reducing the amount of calculation that needs to be performed. We proceed, as in [11], by first applying Razavy's method to a molecule with discrete bound and unbound states, and then evaluating those results in the limit of continuous unbound states. In chapter 6 the results of the numerical calculation of the reflection and transmission coefficients are given. Specifically, we consider the case of a molecule with a double well binding potential with a single bound state and continuous unbound states incident upon a delta barrier. The relative motion eigenstates are defined, and the numerical method used to solve the equations derived in chapter 5 is outlined. The numerical results are then presented. We present plots of the probabilities of reflection and transmission in the bound and unbound states as functions of the molecular CM wave number, i.e. the CM momentum. We find that the probabilities of reflection and transmission in the bound state decrease with decreased binding strength, while the probabilities of reflection and transmission in an unbound state increase with decreased binding strength. In chapter 7, we present the results of a numerical simulation of timedependent molecular tunnelling. We examine the tunnelling of a molecule bound by a double well potential with a single bound state and three discrete unbound states. The numerical method is briefly outlined, followed by the numerical results. We use these results both to give a qualitative description 9 of the tunnelling of a molecule and to determine the trends of reflection and transmission probabilities with respect to molecular wave number. We find that in addition to reflection and transmission, the molecule may also straddle the barrier. In chapter 8, we give a summary of this thesis. We present physical explanations of our results, and discuss possible avenues of future work based on the work in this thesis. 10 Chapter 2 Formulation of the Problem Consider a homogeneous, diatomic molecule incident upon a potential barrier. The atoms of the molecule, each of mass m, have coordinates x\ and x2. The external potential barrier is described by the function V(x), and the binding potential of the molecule is described by UQ(X). The Hamiltonian for this system is given by 2m d2 dx\ d2 + V{x1) + V{x2) + U0{xl-x2). dx2 (2.1) Since the binding potential UQ{X\ — x2) in (2.1) depends explicitly on the relative separation of the molecular atoms, it is convenient to write the Hamiltonian in terms of centre of mass (CM) and relative coordinates. Define the CM coordinate by * . ^ 11 , (2.2) and the relative coordinate by S = x l - x2. (2.3) Then the Hamiltonian may be rewritten as Having determined the appropriate form of the Hamiltonian, we next formulate the problem for two different cases: the time-independent case and the time-dependent case. 2.1 The Time-independent Case Previous studies in molecular tunnelling have employed a time-independent formulation. That is, they have formulated the problem in terms of a timeindependent Schrodinger eqaution. One of the consequences of formulating the problem in this manner is that the solutions which result are steady-state, extended waves. It might seem strange to formulate the problem of molecular tunnelling in such a manner. We do not think of molecules as being extended, nor do we regard tunnelling as a steady-state phenomenon. The time-independent formulation does, however, capture all of the semi-classical processes by which a molecule undergoes reflection from and transmission through a potential barrier. Because of this, the time-independent formu12 lation is able to predict the results of a tunnelling experiment with many trials. That is, if one were to perform many molecular tunnelling experiments, and were then to calculate from the results of these experiments the average probability of the molecule being reflected and transmitted in the bound state and in an unbound state, one would find that those averages are predicted by the solutions obtained using the time-independent formulation. For this reason, the time-independent formulation is useful not only for experiments with many trials, but also in experiments involving a flux of molecules through a barrier, i.e. the sending of many molecules in a single experiment. We wish to find a solution to the Schrodinger equation HV(x,Z) = E*(x,Z). (2.5) The Hamiltonian operator H is given by equation (2.4), and the quantity E is the total energy of the molecule. Assuming that the molecule has a set of discrete, bound, relative motion eigenstates and a continuum of unbound relative motion eigenstates, the solution to (2.5) may be written in the following form: *(*.0 = EXn(OV'n^) + fdq X,(0'^(^)- (2-6) J n The functions Xn{Q and X?(0 describe the relative motion eigenstates of the molecule. The subscripts n and q label, respectively, the bound and 13 unbound states of the molecule. Substituting (2.6) into (2.5) and applying the Hamiltonian (2.4) explicitly yields ft2 ^ Am ^ -T,2 ^Xn(Q + £x»(OV'u(z) v i x + l)+v(x-l dhl>n(x) dx2 + EXn(0^n(x)C/0(0 + ^fdq Am, + / dq Xq(OM*) Vl* + Ti2 r m ^dxj ^2 X , ( 0 l)+v(*-l d2v (£) d v M*)-—^ + / d(i x,(O^W(0 d{2 = E £Xn(OV ; n(^) + / dq X(j(0^?C Equation (2.7) may be significantly simplified. and Xq(0 are First, the functions (2.7) Xn(0 the relative motion eigenstates, and thus form a complete, orthonormal set of solutions to the relative motion Schrodinger equation. That is, Xn{0 and Xq(0 must respectively be solutions to the equations nzd2Xn(o m de £'o(OXn(0=*nXn(0> (2.8) + f/o(Ox,(0 = e,x,(0- (2.9) + and -n2 d2Xq(0 m dt;2 The terms en in (2.8) refer to the bound relative motion energy eigenvalues, 14 and thus e„, < 0 for all n. The terms eq in (2.9) are the unbound relative motion energy eigenvalues, and thus eq > 0. If we define V (x-,£) = V ( x + I ) + V {x ~ I ) ' t n e n ( 2 - 7 ) m a y b e rewritten using (2.8) and (2.9), yielding 4^" Z , ^ 2 *"(0 + L en^n(--c)Xn(0 + EXn(0Vv(-''-)V'(.T,0 + / * / e,'0,(.r)x,(O + ~ / <*? ^ r ^ x , ( 0 + / dq Xq(S)Mx)V(x, 0 = E TsXniOtPnix) + dq Xq{0^q{X) (2-10) The equation given in (2.10), in turn, may be reduced to a pair of coupled differential equations, one corresponding to the molecule in a bound state, and one corresponding to a molecule in an unbound state. The derivation of the former is shown below; the derivation for the latter is similar. The relative motion eigenstates form a complete, orthonormal set of functions. We take advantage of this by multiplying both sides of (2.10) by -$rXm(£)i integrating over all £, and completing the sums over n and inte- grations over q. Since oo / d£ Xl(OXn(0 -oo 15 = Smn, (2.11) where Smn is the Kroenecker delta, and dt xr„(Ox,(o = o, (2.12) (2.10) reduces to d^nOc) + klipn(x) - Y, Zmn(x)ipm(x) - / dq Zq,n{x)ipq,(x) = 0, (2.13) where Zv(x) 4m <% xxoxm V x + V x (2.14) and kl 4m (2.15) 2[E Equation (2.13) is referred to as a multichannel Schrodinger equation. The "channels" in the multichannel Schrodinger equation are the various relative motion energy states of the molecule. The function Zlw[x) is referred to as the effective potential for the molecule. The effective potential is a function which captures the effect that both the external potential V(xj) and the relative motion of the atoms in the molecule have on the centre of mass motion without needing to explicitly write the equations in terms of the relative motion coordinate £. Specifically, the effective potential may be thought of as the potential barrier encountered by a molecule that is 16 incident in state/ "channel" \x and reflected or transmitted in state/ "channel" v. Throughout this thesis, we will follow the convention of using Greek letters to refer to both bound and unbound states. When referring specifically to bound states, we will use the indices (m.n); when referring to unbound states, we use (q,q')- We note, finally, that in (2.13) the labels m and n have been exchanged, and all labels q have been replaced by q'. To derive the corresponding equation for unbound molecular states, we multiply both sides of (2.7) by -$rXq'{0 an d perform a similar derivation as the one shown above, yielding ^ - ^ ox + kfoq(x) - Y, Zmq{x)^m{x) m - I dq' Zq,q{x)iW{x) J = 0. (2.16) Note that we have switched the labels q and q'. In chapter 3, we use Green's function methods to obtain a set of formal solutions to (2.13) and (2.16). We then use the formal solutions derived therein to define expressions for the probabilities of reflection and transmission in the bound state and in an unbound state. In chapter 4, we present analytical results obtained using the formal solutions presented in chapter 3. In chapter 5, we outline a method for solving the formal solutions numerically using the method of variable reflection and transmission amplitudes. In chapter 6, we present the results of the numerical method outlined in chapter 5. As a final note, we point out that in the special case where all molecular 17 states are discrete, (2.13) and (2.16) reduce to f^M + klMx) -Y^Z,u{x)^{x) = 0. (2.17) This equation, as indicated by the use of Greek letters as channel labels, is valid for both bound and unbound states. Discrete unbound states in a molecule occur as a result of the molecule having a finite range of interaction, L. It was found in previous work [11] that, in order to obtain the probabilities of reflection and transmission for a molecule with continuous unbound states, it was necessary to first derive probabilities of reflection and transmission for the case of a molecule with finite interaction range, i.e. one with discrete unbound states. The corresponding probabilities for a molecule with a continuum of unbound states were then obtained by taking the L —• oo limit of the probability expressions corresponding to a molecule with discrete unbound states. We will follow a similar method in deriving the probability expressions in chapters 3 and 5. We next derive a set of equations for the time-dependent case. 2.2 The Time-dependent Case In addition to the time-independent formulation outlined above, this thesis will also consider a time-dependent formulation. This differs from the timeindependent formulation, most obviously, in that it considers the behaviour of 18 the molecule as a function of time, whereas the time-independent formulation is concerned with steady-state solutions. One consequence of this is that we may examine what we would expect the molecule would look like (i.e. its predicted probability distribution), as a function of time as the molecule is interacting with the barrier. As well, while the time-independent formulation is concerned with extended waves, the time-dependent formulation considers the molecule as being localized in space, in the form of a wave packet. As a result, we expect to obtain different results for probabilities of reflection and transmission in the bound and unbound states using the time-dependent formulation than we would using the time-independent formulation. For the time-dependent case, we wish to solve the Schrodinger equation H^(x^,t) d'iHr. £ t) = th°i{d^T), (2.18) with H defined as in (2.4). For this case, we are only interested in discrete bound and unbound states. The reason for this arises from the complexity of the numerical simulations involved in modelling a tunnelling molecule. We found that, in order to obtain results in a reasonable amount of time, we were limited to a molecule with four states (one bound, three unbound). By contrast, in previous investigations of the molecule with continuous unbound states using the time-independent formulation [11], the continuum of unbound energy levels was typically approximated by as many as forty-nine test points. If in this work we attempted to model the continuum using 19 three unbound test points, the spacing between energy levels would be so large as to render the approximation useless. Thus, rather than attempting to approximate the continuum with such a small number of test points, we instead choose to investigate the case of discrete unbound states. We assume (2.18) has a solution of the form *(z,£,0 = £ x , ( £ M ' M ) - (2.19) V In other words, we choose the time dependence of the molecule to be captured entirely in ipu(x, t). This means that, as in the time-independent case, the functions Xv{Q form a complete, orthonormal set of solutions to the Schrodinger equations (2.8) and (2.9). This, in turn, allows us to follow similar steps as those used in the previous section in deriving a time-dependent multichannel Schrodinger equation, d2ip„(x,t) — 2 K,urpv(x, 0 - L Z+vWM**f) . di>u{x,t) = -*7 gl , (2.20) where 4 = f?e,, 7S 4777/ T' (2-21) (2 22) - and Z^v{x) is defined as in equation (2.14). In chapter 7, we will outline a numerical method used to solve (2.20), as 20 well as the results of that numerical computation. 2.3 Summary In this chapter, we formulated the problem of a molecule incident upon a potential barrier. We considered both time-independent and time-dependent formulations. For both formulations, we derived multichannel Schrodinger equations. These equations will be solved in the following chapters. 21 Chapter 3 Formal Solution to the Time-independent Multichannel Schrodinger Equation In this chapter, we present a formal solution to the time-independent multichannel Schrodinger equation derived in chapter 2. We show that our solution is essentially a multichannel variant of the well known Lippmann-Schwinger equation used in the study of single particle scattering problems. From the formal solution, we derive expressions for the reflection and transmission amplitudes. Finally, we present expressions for the probabilities of reflection and transmission in terms of the reflection and transmission amplitudes. 22 3.1 The Lippmann-Schwinger Equation Consider a single particle of mass m and total energy E > 0 incident upon a potential barrier V(x). The Schrodinger equation for the particle wave function ip(x) is given by <2y > + k2^(x) dx = \ h „V } i>(x), (3.1) where k2 = - ^ - . (3.2) If we treat the particle as incident from x = - c o , and if we assume that V[x) —> 0 as \x\ —> oo (that is, that ^(x) behaves like a free wave far from the barrier), then we have wave-like boundary conditions for tp(x) as x —> —oo and x —•> oo. A s x - > —oo, ip(x) satisfies ip(x) - • eifex + i?e- Te?JcT. (3.4) and as x —> oo, The elA:;,: term in (3.3) represents the incident wave, and the coefficients R in (3.3) and T in (3.4) are, respectively, the reflection and transmission amplitudes of the wave scattered from the potential V(x). The reflection and transmission amplitudes are related, respectively, to the probabilities of the 23 particle being reflected from the barrier or being transmitted past it. For complicated potential barriers V(x), it is helpful to write the solution ip(x) to the Schrodinger equation (3.1), subject to boundary conditions (3.3) and (3.4), as # x ) = e"* + ± /_~ e ifc l—'I i ^ ^ 1 ) #X'KT'. (3.5) Equation (3.5) is known as a Lippmann-Schwinger equation, 1 and it is obtained by solving (3.1) by means of Green's function methods. The derivation of the three-dimensional version of this equation is well known, and is covered in many texts on mathematical physics (for example, refer to reference [17], p. 411.) The one-dimensional version shown here is derived by similar means. It may be easily shown through direct differentiation that equation (3.5) is a solution to (3.1). It may also be shown that (3.5) satisfies the boundary conditions (3.3) and (3.4), with R and T being given by and We next show how the Lippmann-Schwinger equation may be applied to the 1 Strictly speaking, there exist two Lippmann-Schwinger equations satisfying (3.1) subject to boundary conditions (3.3) and (3.4). The equation shown in (3.5) describes waves scattered away from the barrier V(x); the second equation, describing waves scattered toward V(x), has no meaning in the scenario, so we ignore it. 24 case of molecular tunnelling. 3.2 The Formal Solution to the Multichannel Schrodinger Equation Recall that in the previous chapter, we showed that the wave function of a molecule incident upon a potential barrier in state /x could be expressed as /-co **(*. 0 = E Xn(OVv(•<<•) + / M xq(M>M- (3.8) Note that we have added the label \x to ^ ( . x , £) and ip^(x) in (3.8) to indicate the incident state of the molecule. While the wave function \&/t(x, £) captures all possible reflections and transmissions of a molecule, it is still necessary to specify the initial state. That is, the physics of a molecule reflected or transmitted in state v and incident in state \x will be different from that of a molecule reflected or transmitted in state v and incident in another state ft,'. The functions Xn(0 an d XgCO a r e the relative motion eigenstates, and the functions VVm(x) a n d V v ( x ) are functions satisfying the multichannel Schrodinger equations, d2ij)im(x) dx2 /•oo -kfynnix) - Y, Zmn{x)ij)^m{x) 25 - / d,q Zqlm(x)4'M,(x) = 0, (3.9) and d2ipliq(x) dx2 k2q^m{x)-^JZmq{x)i}iim{x)- / dq' Zq,q(x)ipM,(x) = 0, (3.10) o where the functions Zv^[x) are defined by (2.14) and k2 is defined by (2.15). It would seem that the next step would be to derive formal solutions to (3.9) and (3.10) and from those define expressions for the reflection and transmission probabilities. However, in previous work [11] it has been found that another, less obvious approach is more effective in obtaining the probability expressions. Specifically, we define the formal solution for a molecule with a finite interaction range L (and thus, discrete unbound states), obtain probability expressions for this molecule, and then take the L —> oo limit of those expressions. Recall from chapter 2 that in the case of a molecule with discrete unbound states, the multichannel Schrodinger equations simplifies to ^ ^ + kl^v{x) - £ Z^{x)^{x) = 0. (3.11) This equation applies to both bound and unbound states. Next, consider (3.1). Notice that if we replace i\){x) with VvO*-) a n ^ c n o o s e the function V(x) to satisfy ^ ^ V V ( * ) = E ZUx)^(x), (3.12) then we obtain (3.11). We already stated that the Lippmann-Schwinger equation (3.5) is a formal solution to (3.1). 26 Thus, by making the same substitutions mentioned above into (3.5), we may obtain a similar formal solution to (3.11), 1 /'°° kvX ^x~* —oo, VV(^) - eikvX^ + R^e-ikuX, (3.14) where R,w is the reflection coefficient for a molecule incident in state /i and reflected in state v. As x —> oo, VvW -* T^e*"*, (3.15) where T^w is the transmission coefficient for a molecule incident in state \i and transmitted in state v. Comparing (3.14) and (3.15) to (3.13), we see that the formal solution does indeed satisfy these boundary conditions. Consider (3.13) as x —y - c o . In this limit, \x — x'\ —> —{x — x') for all x'. Thus, (3.13) 27 may be rewritten as W„(x) = eik»x8p, + e-*" T f - * - Yl f^ \llKv elk-x'Z^(x')^(x')dx') ^ J-co . (3.16) I The term in parentheses is integrated over all x' and has no dependence on x, and is thus effectively a constant dependent only on fi and v. Comparing (3.16) to (3.14), we see that the formal solution does indeed satisfy the boundary conditions for x —> - c o , and that R^ may be defined by R ^ = ^7 e**"1 Z^'yiU^'W- E / (3-17) Likewise, in the a; —» oo limit, |x — x'| —> (a; — x') for all x', and thus (3.13) may be written oo boundary conditions, and the transmission amplitude T^ may be defined as 1 V = 5,„ + — - £ i r°° / ; e" lfc ^ Z^{x')^v{x!)dx'. (3.19) Having defined the reflection and transmission amplitudes, we may calculate the probabilities of reflection and transmission. 28 Before moving on to that calculation, however, we point out that the formal solution (3.13) may be extended to the case of a molecule, incident in a bound state, with continuous unbound states. For this thesis, we only consider the expressions corresponding to a molecule incident in a bound state i. Writing out (3.13) explicitly in terms of bound and unbound states, and then changing the sums over unbound states into integrals over q', we obtain two equations: 1 ZlKn roo roo d( + ^T Zlfhji i7 J— OO m J-oo etk^-*\Zq,n(x')TPw(x')dx', (3.20) J— OO for a molecule reflected/transmitted in a bound state, and Mx) eik"\x-x'\Znui(x')i>im(x')dx' = ^ T E f°° + ^T f°° dl' r ZlKq ./—oo ^k^-x'\Zqlq^)^q,{x')dx\ (3.21) •/—oo for the molecule reflected and transmitted in an unbound state. Following steps similar to the ones shown above, we may define the coefficients of reflection and transmission in a bound state, 1 /-co R e ™ = WT- £ / Z%Kn 1 + ^T ZtrZ^ m Zmn(x')lPlm(x')dx' J— oo roo d( roo l' Jo e^Z^WPiA^W, J — oo 29 (3-22) and 1 T- = 5 • J u in — m /-oo e~lknX — £ / 00 > ~ .'K*n m J- Zlroo roo 1 + ^ r / e-^xZnq,(x')iJjw(x')dx'., mi(x')dx' (3.23) J—oo and in an unbound state, /"OO 1 ^ =^ £ / "17 n i J roo 2lkq e nX Zmq{x!)Am(x')dx' roo oo / -00 e ^ Z ^ ^ C * ' ) ^ ' , .1 JO (3-24) -00 and 1 /ilKq f°° m — / 2ifc ,z/c„v .;o Jo 1 / J —00 e^k«xZq,q(x'yi/jiq,(x')dx'. dq' (3.25) J—00 The reflection and transmission coefficients defined above will be put to use in chapter 4 in obtaining the reflection and transmission coefficients for a molecule incident in the bound state and reflected/transmitted in the bound state. We next show how to derive expressions for the probabilities of reflection and transmission. 30 3.3 Calculation of Probabilities To obtain reflection and transmission probabilities, we must first calculate the incident, reflected, and transmitted probability fluxes at infinity. From the fluxes, we may derive the probabilities of reflection and transmission in a given state by making use of flux conservation. The wave function tyfl(x,£) for a molecule with discrete unbound states incident in state fi is given by (3-26) *M(^0 = E X * ( 0 ^ ( 4 We are interested in the behaviour of the wave function as x —> ±00. From the boundary condition (3.14), we find that as x —> —00, * , ( * , 0 = E x m (eifc*x 00, M * . 0 = £ x * ( 0 [T^e-^**) • (3.28) To calculate the probability amplitude associated with a molecule reflected in state is, we take inner product of the ket vector of the molecular wave function, [i'^x, £)), and the bra vector of the relative motion eigenstate 31 for channel v, {Xv(0\: J

+ Rlt4,e- * ) - eikfiX + R^e 5v4> lk,/X (3.29) . Equation (3.29) describes a combination of two waves. The etk^x term represents the incoming molecule in state [i\ the wave R^ve~%kvX represents the reflected molecule in state v. Recall the definition of probability flux for a wave function ip(x): j(*) m{x) n nx)^-^)^ 2iM dx M dx rid dip(x) dx , (3.30) where ^s(z) is the imaginary part of z, and M is the mass (in this case the molecular mass, M = 2m). The flux of the incident wave is then zm L J (3.31) Zvi and the flux of the reflected wave is JRn» = 7^—3 [{R^e -ikvx\* IKV) tifiuB — iku'X\ I11*-(iv R 2 ""V ~2m' (3.32) The probability of reflection, with state transition /./, —> //, is obtained 32 from the incident and reflected fluxes by means of the following equation, jRliv VRnu (3.33) Jfj, Substituting (3.31) and (3.32) into (3.33) gives — _ l ff I (3.34) Taking the x —> oo limit of (3.26) and following similar calculations shows that Ky \rp |2 (3.35) Kit We are interested in the total probability of reflection and transmission in a bound and unbound state. To find the total probability of reflection in a bound state, one may simply add up all the probabilities of reflection for all bound states m for a molecule incident in state f.i: (H) V-^ V ^ ""m I p VRb — /_^ PR/im — l_! ~TT I '""• m (3.36) 'V By similar reasoning, we find the total probability of transmission in a bound state to be: I nw _ V — IT Prb ~ z_^ u Il /'•">- I Similarly, we obtain the probabilities of reflection and transmission in an 33 unbound state by summing the probabilities for all unbound states q'\ PRu E^I'WI2. q' (3-38) 'Ll and tilE^ITWI2- (3.39) We now wish to obtain expressions for the unbound probabilities for a molecule with continuous unbound states. As the interaction range becomes infinite, the spacing between unbound wave numbers approaches TT/L. That is, the allowed unbound wave numbers become continuous in the limit of infinite interaction range. It is shown in [11] that because of the TT/L spacing between unbound wave numbers in the limit L —» oo, the sums given in (3.38) and (3.39) transform into the integrals Pn^lfdc/^IR^2, (3.40) Pri=l-jdq'kf\Tm,\\ (3.41) and in the limit of continuous unbound states. We now have expressions for the probabilities of reflection and transmission in a bound state and in an unbound state, in terms of the reflection and transmission coefficients R^ and Ttw. 34 3.4 Summary In this chapter we found a formal solution to the multichannel Schrodinger equation. From this solution, we derived expressions for the reflection and transmission amplitudes. We then derived expressions for reflection and transmission probabilities in terms of the respective amplitudes. 35 Chapter 4 Preliminary Analytical Results In this chapter, we consider the special case of a molecule with a delta binding potential of arbitrarily weak strength. Using this as a representation of the potential for the weakly bound molecule, we derive three analytical results. The first is that the bound state of an arbitrarily weakly bound molecule incident upon an infinite barrier ceases to exist when the centre of mass is far from the barrier. We also show that, far from being the easy problem as we assumed it would be, the rigorous calculation of the reflection probabilities is quite difficult. However, we obtain an asymptotically valid approximation to the molecular wave function in spite of these difficulties. Finally, we determine the probability amplitudes for a molecule incident in a bound state and reflected or transmitted in a bound state for the case where the external potential is a delta barrier. 36 4.1 A Special Case: The Delta Well Binding Potential Recall from Chapter 2 that we expressed the wave function of a molecule incident upon a potential barrier as /•oo * , i M = £Xu(Oi/Vn(x)+/ dqXlM)^{x), (4.1) where the functions Xn{0 and Xq{0 describe the relative motion of the atoms in a molecule subject to a binding potential £/ 0 (0- These functions, by definition, are solutions to the relative motion Schrodinger equations -ft2 d2y (£) — J 7 ^ 1 + W X n t f ) = *nXn(0, (4-2) d^A m for bound states (e„ < 0) and -ft2 d2\ — ^ (() ^ + U0(OXq(O = e,X,(0, (4-3) for unbound states (eq > 0). For our analytical work, we consider the case where the binding potential £/o(0 is a delta well potential, f/o(0 = ~hS(0, 37 (4-4) where A;, is the binding strength of the potential well, in units of energy multiplied by length. In previous work [5, 6, 11, 12, 13], the molecular binding potential has been modelled as a double square well potential with a central square barrier. Indeed, the double well potential is the binding potential that we will use in our numerical work, which we discuss in chapters 5 and 6. However, the complexity of the relative motion eigenstates for this type of potential well makes it difficult to work with analytically. Thus, as a representation of a weak double square well potential, we use the delta potential defined in (4.4). Even though such a potential is much simpler, it still captures many of the main physical features of the double well potential. Crucially, for a delta well, there exists only one bound relative motion eigenstate, Xo(0> a s w e W1U show later. This is an important feature since, in one dimension, a molecule bound by a potential well of any form has at least one bound state, provided that the potential is finite everywhere and has at least one well. We will also show later in this chapter that many of the effective potentials Z$v{x) have a very simple dependence on the binding potential strength \ . We will use this simplicity to analytically calculate the probabilities of the molecule being reflected or transmitted in the bound state. We consider two cases: that of a molecule with finite interaction range L, and that of a molecule with infinite interaction range. The latter case will be investigated in section 4.1.2, and will be of use in section 4.3, where we attempt to find expressions for the probabilities of the molecule reflecting 38 or transmitting in the bound state when incident upon a delta barrier. The former case will be of use in section 4.2, when we consider a weakly bound molecule incident upon an infinite barrier. Specifically, we consider the case of a molecule whose interaction range, L, has a linear dependence on the centre of mass coordinate, x. The dependence of the relative motion eigenvalues on L is shown in section 4.1.1. 4.1.1 Delta Well Potential with Finite Interaction Range For a molecule with a finite interaction range L, we model the binding potential as an infinite well of length 2L centred at £ = 0, with a delta well as defined in (4.4) at £ = 0: Uo(S) = { (4-5) For this interaction potential, the separation of the atoms in the molecule is never greater than L. We also have the boundary conditions Xn(0 = 0 and X 9 (0 = 0 for |£| > L. In addition, the delta well imposes discontinuities of the ^-derivatives of Xn(0 and x ? ( 0 a t £ = 0: d Xo(0\ <*Xo(0| 39 - ~mX\< (n\ (A a) for the bound state, and d dxq(0\ d£ k-o+ Xq{Q I df k-o- —^r-X?(0), (4.7) for the unbound states. The boundary conditions defined above allow us to obtain exact solutions to (4.2) and (4.3) for the potential well (4.5). Doing so, we obtain o, Xo(0 = A 0 sinh(a L [L + ^]), -L<£<0, i40sinh(aL[L-£]), 0 < { < L, (4.8) 0, for the bound state, and x,(0 = o, i<-L, Aqsm(qL[L + S}), -L<£<0, Aqsm(qL[L-Z]), 0 < £ < L, (4.9) 0, for the unbound states. The quantities &L and qi are the allowed relative motion wave numbers for a given interaction range L, while A0 and Aq are the normalization coefficients. The former are given by the equations aL = -me0 m\b 40 w M (4.10) and [rneq m,Xb QL = \l-jTT = -^2 tan{qLL). (4.11) Recall that c0 and tq are, respectively, the bound and unbound relative motion energy eigenvalues. The quantity «£, must satisfy (4.10) in order for the relative motion eigenstate (4.8) to satisfy the discontinuity condition (4.6). Likewise, q^ must satisfy (4.11) in order for (4.9) to satisfy the discontinuity condition (4.7). We choose the coefficients OLI and qi to be positive. Thus, equation (4.10) indicates that there exists, at most, only one bound state, since there exists at most only one positive solution a^ for any value of L. Indeed, there exists a critical value for L below which there is no bound state. This will be explained further in section 4.2. The periodic tangent function in (4.11), on the other hand, indicates the existence of an infinite number of discrete unbound solutions, since there are an infinite number of positive solutions to (4.11). The amplitudes A0 and Aq are given in terms of OLL and q^ as follows: A0 = VaL . /sinh(ai£.L) cosh(aLL) (4.12) — CXLL and Aq = , /Tq *; = . 2qLL - sm(2qLL) v (4.13) In this section, we have considered the case of a molecule with a constant, 41 finite interaction range L. In section 4.2, we will modify these equations to describe the relative motion eigenstates and energies of a molecule with a variable interaction range dependent on the molecule's centre of mass coordinate, x. 4.1.2 Delta Well Potential with Infinite Interaction Range For a molecule with infinite interaction range, there are no infinite barriers at ^ = ±L, and the binding potential is given by (4.4). We consider the bound state first. Solving equation (4.2) for the potential (4.4). we find, after imposing the boundary condition Xo(0 -> 0 as |£| —> oo and normalizing, Xo(0 ae°*, £ < 0, Q £ > 0, v^e- «, (4.14) where mXb a 2ft2 (4.15) Note that this value of a is what we obtain if we take the L —> oo limit of equation (4.10). Recalling from equation (4.10) that the binding energy e0 is defined in terms of a by the equation e0 ~-\f-, we may use (4.15) to write eo as 4Ti2 42 (4.16) For the unbound states, we use symmetry to obtain the behaviour for £ —» ±00. For a molecule with a symmetric binding potential, the relative motion eigenstates must have either even or odd symmetry. We will show, later in this chapter, that for a molecule with a symmetric binding potential incident upon a delta barrier, the effective potentials corresponding to transitions between relative motion states of opposite parity vanish. It may also be shown [11] that, to satisfy conservation of parity, a molecule with relative motion states of definite parity may only transition to states of the same parity (i.e. even —* even or odd —> odd). Since the bound eigenstate has even symmetry, and since the binding potential (4.4) is symmetric in £ (i.e. relative motion eigenstates have definite parity), the molecule will only make transitions to even unbound states, and thus the odd relative motion eigenstates can be ignored. Therefore, we choose the unbound states to have even symmetry. Relying again on the derivative discontinuity condition (4.7), we find ert + be-'"*, £<0, X,(0 = (4.17) bej^ + e-1"^ £ > 0, where the coefficient b is given by iq + a (4.18) The unbound wave number, q, is continuous and positive (i.e. all positive q 43 values are allowed) and is related to the relative energy eq by (419) 5V. m These equations for the relative motion eigenstates and energies of a molecule with continuous unbound states will be used in section 4.3 in expressing the effective potentials for a molecule incident upon a delta barrier. We now proceed to our analytical results for the molecule incident upon the infinite barrier. 4.2 Molecule Incident Upon an Infinite Barrier As part of our analytical work, we consider the seemingly simple case of a weakly bound molecule, with a binding potential given by (4.4), incident upon an infinite barrier, 0, x < 0, oo, x > 0. V(x) = { (4.20) It is reasonable to expect that it will be easy to calculate the reflection amplitudes for a molecule incident upon this barrier as the molecule becomes very weakly bound. By physical intuition, we expect the molecule to have increasingly greater probability of breaking up and reflecting in an unbound 44 state as the binding energy goes to zero. That is, we expect the probability of reflection in a bound state to approach zero, and the probability of reflection in an unbound state to approach unity, as Aj —» 0 + . Actually calculating the reflection coefficients, however, proves difficult. The infinite barrier presents a problem if we wish to define the relative motion eigenstate as we did in (4.1), that is, as a function dependent only on the relative separation of the atoms, £. To see why, recall the definitions of x and £ from chapter 2: * ^ , (4.2!) 4 = X! - x2. (4.22) S and From these, the atomic coordinates x\ and .r2 are, xx = x + \t X2 = X (4.23) k- The infinite barrier imposes the requirements x,\ < 0 and x2 < 0. From this, we conclude that, for a molecule incident upon an infinite barrier, 2x < £ < -2.x. (4.24) The placement of the minus sign in (4.24) is a consequence of having x < 0, i.e. the CM is always confined to x < 0. This is equivalent to rewriting the 45 binding potential function of x and £, U0(x,O = |£| > - 2 x , oo, -A 6 — 2x, we can use the eigenvalue and eigenstate equations of Section 4.1.1 to define the relative motion eigenstates x.o(x-,Q and Xq{xiO °f a molecule with a delta well potential incident upon the infinite barrier: Xo(x^) ={ x,M) = 0; £ < 2x. j4 0 (x)smh(a(x)[£-2x]), 2x < £ < 0, -A0(x) 0 < £ < -2x, sinh(a(x)[£ + 2x]) 0, - 2 x < £, 0, ^ < 2x, Aq(x)sm(q{x)[£ - 2x]), 2x < £ < 0, -/l g (x)sin(<7(x)[£ + 2x]), 0 < { < -2x, 0, - 2 x < £. (4.26) (4.27) The quantities a(x), q(x), AQ(x), and Aq(x) in (4.26) and (4.27) are defined 46 as follows (recall that x < 0): , s mX>, a[x) = — 2h - 2 tanh[2xa(x)], (4.28) mAfi q(x) = —^£tfm[2xq{x)}, (4.29) Ja(x) A0(x) = V = , '2xa(.x) — sinh[2x«(x)] cosh[2xa(.x)] (4.30) and Aq{x) = J2q{x) V . (4.31) ysin[4:r(:7(.x)] — 4xq(x) Equation (4.28) has significant implications for the behaviour of the molecule as the binding strength A^ —> 0 + . This equation not only defines the satisfactory value of a(x), but states the conditions under which a satisfactory value exists at all. To understand how this is so, it helps to consider a(x) as an independent variable, and to consider the left and right sides of (4.28) as separate, a(x) dependent functions, fihs[a(x)] = a{x) (4.32) and frhs[a(x)],= m\, -—2-tanh[2xa(x)], (4.33) for a given value of x (see figure 4.1). The positive value of a(x) which satisfies (4.28) is that value at which the two functions intersect. But if one 47 examines plots of fuls[a(x)] and /,-/,,.,[a(x)] with respect to a(x), it becomes clear that such a solution only exists if dfihs(x) dfrhs(x)\ < da(x) \a(x)=o da(x) Mz)=o' (4.34) that is, if lim a(x)->0+ ^ = 1 da(x) d lim a(x)-^o+ da(x) -mXb = lim < x \2x + 2 dx — TtlAl) —T-g— tanh[2xa(.x)J cosh~2[2.Ttt(x)] (4.35) da(i a(x) By using (4.32) and (4.33) to express x as a function of a(x), it may be shown that xa(x) 0 and ~pr —> 0 as a(x) —> 0. Thus, in the limit a.(x) —> 0, (4.35) reduces to 1 < ~mXbx V (4.36) Rewriting (4.36) in terms of x, we get 2 -n x < mXi, (4.37) But (4.37) implies that the bound state of the molecule incident from x = —oo may only exist so long as the centre of mass of the molecule does not 48 1.6 1.4 1.2 1.0 0.8 0.6 -\ 0.4 0.2 lhs a(x)] = i(x) x = -1 x = -0.5 X = -0.3333 f 0.0 — —i 0.0 1 1 1 r 0.2 0.4 0.6 O.I 1.0 1.2 1.4 1.6 a(x) Figure 4.1: Plots of j\hs{a{x)) and frhs{a(x)) for various values of x. We have chosen units such that xmax = —0.5. For x < xmax, a(x) and ///i 5 (a(:r)) intersect for a positive value of a(x). For x > xmax they do not. pass a certain maximum value of x. We define this maximum value, xmax, •£• max. — m,\h If the CM of the molecule comes any nearer to the barrier than xmax, as (4.38) the bound state of the molecule will no longer exist. This means that a molecule, incident upon the infinite barrier in the bound state, will break up into 49 separate atoms if its CM passes the critical point xmax. Consider what happens as the molecule becomes more and more weakly bound, i.e. as A& —• 0 + . From equation (4.38), we find that, as the molecule becomes arbitrarily weakly bound, the distance from the barrier at which the molecule breaks up becomes arbitrarily large. In other words, an arbitrarily weakly bound molecule breaks up arbitrarily far from a,n infinite barrier. We would expect the molecule to break up if the CM goes any closer to the barrier (i.e. xmax < x < 0). Another possibility is that the molecule is reflected in the bound state, with the CM never going farther than xmax. We discuss this possibility later in this section. It is important to note that to obtain the above result, we relaxed rigour by temporarily ignoring the implications of replacing Xo(0 a n d Xq(0 with XoC^iO and Xq(x>0- We did this in order to illustrate an important physical point. Indeed, when we do take such considerations into account, we find that the exact calculation of the probability of reflection in an unbound state is extremely difficult, due to the ^-dependence of the relative motion eigenvalues and eigenstates. To illustrate this difficulty, consider a molecular wave function of the form *,x(z>0 = £ x * ( s > 0 V w ( s ) > (4-39) where /i indicates the initial state of the molecule and labels all the discrete, x-dependent relative motion states (one bound state, an infinite number of 50 unbound states). If we apply a similar process to that used in chapter 2 to obtain the multi-channel Schrodinger equation, we get the following system of equations for x < 0: * ^ + kl(x)^(x) + J2 v ( s where kl(x) Am (E - Y, Z^(x)^(x) + W^(X)I/J^(X) dx = 0 (4.40) — e,y(x)) (E being the total molecular energy, e„(x) being the x-dependent relative motion energy) and the functions Z^(x), u(f)U(x) and a^(.x) are defined by -2s 4m Ztv{. ) = -pi I x Xl{x,0x Integrating (4.44) over £ would result in an even more complicated equation than that given in (4.40). It would seem as though any further analytical work on the subject would be considerably challenging. However, recent work on our part has shown promise in obtaining analytical expressions for the molecular wave function, from which the reflection probabilities may be found. We assume a solution of the form X \ $o(z,OH where xmax X-max > . , (4-45) and Xo(-x>£) are defined as before. In order for $o(- T iO to be a valid solution to the Schrodinger equation, it must satisfy the following two conditions: x lim $ 0 ( x , O = 0, (4.46) lim ^ i l l = 0 . (4.47) • x T17 a x {J X It is clear immediately that (4.45) satisfies (4.46). Our work has revealed that asymptotic expressions for <&o(x,!;) may be obtained for the limits x —> - c o 52 and x —> £„,„_. As x —> — oo, we have $ o ( * , 0 ~* [etfcoa; + Roe-ikoX]e-aM, (4.48) and a s x - > xTOrax , we have -ikox] o ?;fe 0 x *o(x,0 1/2/1 -3 o 1] -a (4.49) where R< _ o2ik0xmax (4.50) and r mXb t \ a = hm alx) = —~-. (4.51) It may be shown that the asymptotic form of $ o ( ^ , 0 given above satisfies the boundary condition (4.47). An interesting consequence of this solution is that it describes a molecule reflected in the bound state. Indeed, both the incident and reflected fluxes are found to be 2 m =m m Tikft ft/cn (4.52) That the fluxes are equal means that the probability of reflection in the bound state is unity. If the wave function given above accurately describes the behaviour of a molecule incident upon an infinite barrier, then this is 53 a most interesting result. We note that this solution is an approximation of asymptotic behaviour. As well, we note that this solution does not take into account transitions between states. Nonetheless, we consider the above results to be a promising beginning to further work. We next consider the case of a molecule incident upon a delta potential barrier. 4.3 Probability of Reflection and Transmission in t h e Bound State for a Molecule Incident in the Bound State Upon a Delta Potential Barrier In this section, we determine the reflection and transmission coefficients for a molecule incident upon a delta barrier in the bound state and reflected or transmitted in the same bound state. We once again consider a molecule with a delta binding potential. 54 4.3.1 Effective Potential for a Molecule of Definite Parity Incident Upon a Delta Barrier Recall, from (2.14), the definition of the effective potential Z$v{x): n V|x+|)+v(x-i J—oo (4.53) For this section, we choose the external potential to be a delta barrier of strength A, so that we may write V ix± £ X8[x± z (4.54) and substitute into (4.53). Performing the integration over £, (4.53) simplifies to Z^{x) = 8mA \x:(-2x)x4>(-2x)+X:(2x)x^2x)]. (4.55) If the binding potential for the molecule is symmetric in £, then the relative motion eigenstates x^i^x) and X4>(2x) will have definite parity, i.e. they will either be even or odd in £. We define the boolean function p/Jr, 0 if x,M) is if XiM) is V,,= 1 55 even, odd. (4.56) The effective potential given in (4.55) may then be written as Z+,{x) = ^ [(1 + (-l)^^)X:(2x)x,(2x)} where we have used X/<(_2x) = , (4.57) (—Vf^x^x). From (4.57) we obtain two important results. First, if pv + p^ is an odd number, i.e. if Xi/(0 a n d X(0 a r e of opposite parity, then the above equation is identically zero, and thus Z^v{x) makes no contribution to the formal solution to V v ( x ) o r to the reflection and transmission probabilities. It is because of this result that we have thus far only considered even unbound relative motion states: our work is concerned with transitions from the single, even bound state to an even unbound state, so only the even-to-even effective potentials make a contribution to the formal solution, and hence to the reflection and transmission coefficients. The second important result we obtain from (4.57) is the effective potential for Xv{Ci a n d x(0 °f the same parity. In this case, (4.57) simplifies to Z*(x) = ^Xl(2x)x*(2x). (4.58) This result applies to all molecules with a symmetric binding potential incident upon a delta barrier. We next use the above results to determine the coefficients of reflection and transmission for a molecule with a delta binding potential. 56 4.3.2 Reflection and Transmission Amplitudes Recall from Chapter 3 that we obtained reflection and transmission amplitudes for a molecule incident in state // and reflected or transmitted in state v. For a molecule bound by a delta well, there exists only one bound state of even parity. Thus, for such a molecule, incident in the bound state, the amplitudes of reflection and transmission in the bound state are given by R ™ = ^ T f°° e%k°Xl'Zoo(x')iP00(x')dx> Z'tfcQ J-co 1 /-oo + ^T roo d( e i' ZIKQ JO Zq,v{x')ijq,0{x')dx\ (4.59) J-oo and e Tm = l + 7TT ZIKQ 1 rco + ^T ZIKQ JO Zm{x')i)m(x')dx' J-CO /•oo d( e'ik°x ZrfQ{x')jpw{x')dx'. t' (4.60) J-OO Recall the relative motion eigenstates for the molecule bound by a delta well, given by (4.14) and (4.17). Substituting these expressions into (4.58) gives the effective potentials Z00(x) and Zq*0(x) from equations (4.59) and (4.60): ZOQ(X) = Aae-ia]x], (4.61) and Zql0(x) = Av/Se- 201 *! (be2iq>W + e" 2 "''!^), 57 (4.62) where 16mA A = ^ ^ , .A „. (4.63) and b is defined by equation (4.18). The effective potentials defined by (4.61) and (4.62) are both proportional to the normalization constant a, which is itself directly proportional to the molecular binding strength coefficient A&, as seen in equation (4.15). Since the effective potentials are finite for all finite a, this means that for an arbitrarily weakly bound molecule (i.e. A& —• 0), the effective potentials Z00(x) and Zoq'(x) both approach zero for all x. To understand how this impacts the reflection and transmission coefficients, we make two assumptions, both on physical grounds. First, since •tpoo(x') and ' 0. Thus, we expect that for a weakly bound molecule the wave functions ^oo( s ') and ipoq'(x') will not have any dependence on a. This means, for example, that the wave functions I/JOQ(X') and ipoq'(x') will not be proportional to I/a. as a —> 0. Hence, as Af, —> 0, the integrands in (4.59) approach zero for all x', i.e. lim A a e - ^ ^ e ' ^ V o o ^ ' ) = 0, a-*0 58 (4.64) and lim k^e-2a\x'\be2iqW + e-2iq\x\)eik°x'^{x') = 0. (4.65) This, in turn, means that the integral terms in the reflection coefficient will vanish, i.e. R00 —> 0 as A;, —• 0. Hence, the probability of an arbitrarily weakly bound molecule, incident in the bound state, reflecting in the bound state goes to zero. This result seems to be in line with what we would expect for a weakly bound molecule. That is, we would expect that a weakly bound molecule would have a greater chance of "breaking up" upon contacting the potential barrier, and thus reflecting or transmitting in an unbound state. However, when we examine (4.60) and apply the same argument, i.e. that the vanishing effective potentials result in vanishing integral terms, we conclude that lim Too = 1. (4.66) Recalling the expression for the probability of transmission given in (3.37), this means that the probability of transmission in a bound state for a molecule incident in the bound state approaches unity. Unfortunately, we cannot apply a similar argument to the expressions for Roq and T0q. The integrals in these equations contain the effective potentials Zqiq(x) which, in the limit a —> 0, are given by Zq,q{x) = 4Acos(2gx) cos(2g'x). 59 (4.67) Thus, the integrands do not approach zero for all x. However, the results for •/7.00 and T00 in the cv —> 0 limit imply that the reflection and transmission probabilities in an unbound state go to zero. We note that, once again, we have relaxed rigour in order to obtain a physically interesting result. More rigorous analysis of the reflection and transmission coefficients was limited clue to the states '0A„,(.x') not being known explicitly. In chapters 5 and 6, we outline a procedure to numerically calculate the reflection and transmission coefficients using the method of variable reflection and transmission amplitudes. We compare the analytical results to the numerical results for a molecule with a single bound state, incident upon a delta barrier in the bound state, as the binding strength becomes weakened. 4.4 Summary We have examined the cases of a molecule with a simple binding potential, the delta well, incident upon an infinite barrier and a delta barrier. We have determined that such a molecule, incident upon an infinite barrier, can exist in a bound state only up to a certain distance away from the barrier. The distance from the barrier at which the bound state ceases to exists becomes arbitrarily large as the molecule becomes arbitrarily weakly bound. We have also found asymptotic expressions for the molecular wave function incident upon the infinite barrier, which as of this writing are open to investigation. 60 Finally, we found that the probabilities of reflection and transmission in the bound state vanish in the limit of arbitrarily weak binding strength for a molecule with a delta well binding potential incident upon a delta barrier. 61 Chapter 5 The M e t h o d of Variable Reflection and Transmission Amplitudes In this chapter, we outline a method, developed by Razavy [1], for obtaining the reflection and transmission amplitudes, i?/(,„ and T,lu, of a molecule incident upon a potential barrier. We do so by first defining the so-called "variable" reflection and transmission amplitudes, R^y) and 7}t„(y), from which R^y and T^ may be obtained. We then derive a set of coupled, nonlinear, first order differential equations which, when solved, give the values of i?,M,y(y) and TM„(y), and hence Rw/ and 7jt,y. We first derive the equations for Rfj,v{y) a n d 7)t[/(y) for the case of a molecule with a finite interaction range L (and, hence, discrete unbound states). Then, we obtain expressions for a 62 molecule with continuous unbound states. We do so by using the expressions for R^(y) and 7)t£/(y) in the discrete case to calculate R,iv{;y) and TIJL/(y) in the limit L —> oo, i.e. a continuum of unbound states. 5.1 Discrete Unbound States Recall from chapter 2 that for a molecule with a single bound state and an infinite number of discrete unbound states, we have a multi-channel Schrodinger equation of the form (dr2 + f c W ^ - " ^ ~ 51 ^"faNwOO = 0 ' (5-1) where Z ^ ( . T ) is the effective potential defined in equation (2.14) and kv is the CM wave number corresponding to state v. We note once again that we use Greek symbols to refer to both bound and unbound states. As explained in chapter 3, we may write a formal solution to (5.1) as follows: VvW = JkvXV + ^ - E f e ! f c " | x " ' T ' 1 Z M * ' ) M X W " d> (5-2) •'~°° As in chapter 3, the reflection and transmission amplitudes are 1 r°° ik x RI1U = ~— Y, / e " Z^{x')xlj^{x')dx\ lih„ ^ J-oo 63 (5.3) 1 roo T^ = S^ + —-Y, MKU e " "x Z^(x')iP^(x')dx'. (5.4) ^ J -CO From (5.3) and (5.4), it would seem that R^ and TjW cannot be evaluated without first directly solving for the tp^x) terms in each equation. However, Razavy has developed a technique, known as the method of variable reflection and transmission amplitudes, by which R^ and T^ may be evaluated indirectly, without having to solve for "0M(.x). This method is outlined in reference [1]. We first define the cutoff potential Z^v(x,y), corresponding to the effec- tive potential Z y. (x-y) (5.6) The formal solution 4>nU(x,y) to the multi-channel Schrodinger equation for the effective potential (5.5) is then •M*, v) = j * " 1 ^ + 7^-Y,r elM'T~x%^0VW^ v)dx'. (5.7) From (5.7), we may derive, as we did for (5.3) and (5.4), the "variable" (i.e. 64 y-dependent) reflection and transmission coefficients, f00 1 RAV) = ^ E Z,IKU , 1%KV Z^(x')^ld>(x', y)dx\ (5.8) Jy 1 TIW(V) = Thus, by solving for R^,v{y) and T^vty), we may obtain the reflection and transmission coefficients R^ and 7)(,y from the y —•> — oo limits of R^iy) We next show how to obtain the differential equations that yield and R^(y); the derivation of the set of differential equations that yield 7)„,(y) is very similar. We first note that, from (5.7) and (5.8), Vv(y>^) may be expressed as W ( y . V) = e i f c , "V + e - ^ i ^ ( y ) . (5.12) This result will be of use later on. In addition to (5.12), we make use of three lemmas, which we will prove before deriving the final result. 65 5.1.1 Lemma I If we define the element Bpl(y) of the square matrix B(y) by the equation -ikuy Y -^jT Y z^MMv, y)BPi(y) = (5.13) then y, we have dil>p»(x,y) dy 1 2ik„ -2^-E^(B-y)^(v)^(y.y)- (5.15) Replacing \i, with /?, multiplying by B,yp(y), and summing both sides over all p, we then obtain E^W) <9y = —e ikux e-iKy Y -^T~ P i m v Y Z^MPAV^ y)B,/P{y) 4, Mx y) z^zr^"'%^ '' + 2ik V rh dy J V 66 Bu,p(y)dx'. (5.16) Comparing the first term of the right-hand side of (5.16) to the definition of Bvip{y) given in (5.13), we see that (5.10) simplifies to + £ 2^" £ / " ^ ' - ^ U A ^ ^ B ^ y ) ^ . (5.17) Comparing (5.17) to (5.7), we see that Y.dijpfyV)Bv.pi:y) = ^{x,y). (5.18) That is, due to the similarity between (5.17) and (5.7), we can conclude that the left-hand side of (5.17), Y,P g Bvip(y), is equivalent to ipu>„(x.y). Multiplying both sides of (5.18) by the inverse matrix element B~^,(y) and summing over all u', we obtain Y,^'u{x,y)B^,{y) y' = Y,^4^EBMy)B-Uy) dj>pv(x,y) dy V ^ 6p" dipnv(x,y) . dy 67 (5.19) By changing labels x —> x', TT —> p, v —> p, and i/' —> t, we arrive at the result we sought to prove, ^"£'V) 5.1.2 =ZM*',V)BJ(V)- (5.20) L e m m a II The inverse matrix element B~^(y) may be explicity written as B; {v) = ? -^rT,z*r(v)Mv>v)p (5-21) P R O O F : Recall the definition of the matrix element Bpo(y) given in (5.13). If we multiply both sides of (5.13) by B~,Hy) and sum over all p, we get e -ik"y E K W * > = -E-27T-£ z »(v)My>y)E^(y)^(y). p t Z "U1/ <£ (5.22) p This, in turn, simplifies to e -ifci/S/ 57,1 (y) = - E - ^ r E Z^(y)A^(y., y)5», e -*„v 2*^ £^,(y)Vv*(y,y). (5.23) (5.24) Exchanging labels v -+ p and t' —> t yields s-/(y) = - ^ - E ^ W M ) 'p 68 (5-25) 5.1.3 Lemma III Our third and final Lemma is as follows: (5.26) P R O O F : Recall (5.8), which defines R^v{y). If we multiply both sides of (5.8) by B~^(y) and sum over /t, we obtain 1 roo ikvx>7.. fxn E RAvWv) = 57T E / ^"s z* £lhv , Y,iji>4(x',y)BPlt(y) Jy dx'. (5.27) The term in square brackets may be simplified using Lemma I, yielding E R,Av)B;i(v) = 2^" E / " ^'Z^{x'f^v) dx'. (5.28) Exchanging labels [i —•» 7 and x' —> x yields . 1 1 roo E^(v)^ (y) = ^ r E / e^xz< Llhu 5.1 A , Jy f^M^l^. (5.29) dy Final Result We now combine Lemmas I, II, and III, as well as (5.12), to obtain the following set of differential equations for i?/t„(j/)(next page): 69 x Y, Z^(y) \e^!' V + e-^Ry^y)] . (5.30) (5.30) P R O O F : We first take the y-derivative of (5.8), yielding i » r..\ _. J v- r „ . u 7 ,^w,v"i>(x'>y)dx> dy«M=^r^'^'> dy 1 kvVZ 2ifc, Y,J *MMy,y)- (5-31) The first term may be simplified using Lemma III, while the second term may be rewritten using (5.12), yielding ~R,Ay) = ER^(y)B;n1(y) - T^T- £ ^'"Z^y) \e^5„4 + e~^R^(y)} . (5.32) Using Lemma II and (5.12) to rewrite the first term, and expressing the second term as a sum over 7, we arrive at 70 - E E 1^-Z+iW-v [^ and 7, this equation may be more neatly written as X £ Z*,(y) [<*»*,„ + e-^»R,„(!,)] . (5.34) 7 Following a similar calculation, we arrive at the following set of differential equations for TliV(y): ^ t o ) = -E^e-*«T rt (») 7 For both sets of equations, we have removed the dependence on ^^(x'^y), and thus have greatly simplified the calculations needed to obtain the probabilities of reflection and transmission. As well, it may be shown, using either parity arguments or asymptotic expansions of (5.30) and (5.35), that i?M„ = 0 and TIJLU = 0 if states \i and v are of opposite parity [?, Ref Shegelski Hnybida et al] Therefore, if the bound state is even, the molecule will only transition to other even states. We next extend (5.30) and (5.35) to the case of continuous unbound states. 71 5.2 Continuous Unbound States The method developed by Razavy considered a molecule with discrete states. His method was later extended to the case of continuous unbound states, which was reported in [11]. The extension of Razavy's method to the continuum is outlined in this section. When attempting to extend (5.30) and (5.35) to the continuum, we encounter difficulty: the Kroenecker delta terms in the two equations are replaced by Dirac delta functions, which are difficult to deal with in numerical calculations. We avoid this difficulty by defining the intermediary functions Uiw(y) and Qlu,(y) as follows: R^(y) T^(y) = 2iKU^{y)el{k"+K)\ = 2iKQilu{y)e^+k^ (5.36) +V (5.37) It may be easily shown from (5.36) and (5.37) that UfW(y) and Q,iv{y) satisfy the boundary conditions lim Ulw{y) = 0, lim Q,w{y) = 0. (5.38) Substituting (5.36) into (5.30), we obtain -rUtw(y) = -i(kfl + kv)U,lv(y) + — — 72 Ztw(y) 1 1 2^V T ™'~ """ 2ik. v -EE^(y)^(y)Wy)' (5-39) The research in this thesis concerns a molecule with a single bound state and infinite unbound states. It is convenient at this point to separate the terms in (5.39) involving the bound state (labelled 0) from those involving unbound states (labelled q' and q")\ — U^iy) = -i(kfl + K)U,w{y) 4/c„fc, •7^rzAy)UoAy) 2ikv -Zov(y)Uf>o(y) -Uou{y)ZQQ{y)Ulja{y) -J2uo»(y)zMy)uiw(y) - J2 Uq»y{y)ZQqn{y)UlxQ{y) l/{y)UM,(y) q- u -i:E My)z«w>(y)u»Ay)q> (5.40) q" We now take the limit of continuous unbound states. For a molecule with 73 finite interaction range L, i.e. with discrete unbound states, the spacing between successive unbound relative motion wave numbers approaches TT/L as L —> co. We mentioned in chapter 3 that this leads to particular expressions for the probability of reflection and transmission in an unbound state in the limit of continuous unbound states. Similarly, as shown in reference [11], we obtain the following differential equations for t/M,,(y) and Q^„(y) in the limit of continuous unbound states: •^-Ullv{y) = -i(klt + 4/c/t/c,, kv)Utw(y) z^(y) — ZllO(y)U0,y(y) lik^ -lV(y)Z 0 o(y)c^o(y) 1 r°° -- / dq'U0l/(y)Zq,0(y)Uliq,(y) n Jo 1 /°° - - / dg"L/g»I/(y)Z0(?»(y)[/^o(y) IT JO 1 f°° -7T~r / Imk,^ Jo 2inku Jo - 2 / 7T J o d( i"ziw'(y)uu(y) TT JO 1 f°° -- / d<^'QfJ.q>l(y)Zq»0(y)U0u{y) 7T JO 1 f°° l 3-2iktly co 2iirkIJ,-e \ 7T2 /'OO /•OO dq'ZM,(y)Qq,„(y) /'OO /'OO / -^\2k0Q0q,(y)f. IX JO (5.46) fc0 In principle, to obtain the probabilities of reflection and transmission for our original molecule (i.e. for the molecule without a cut-off potential) we simply evaluate pRb{y), Pn(y), PRu(y), and pTu(y) in the limit y -> - c o . However, for numerical calculations it is not possible to take y —> —co. Nonetheless, we find that a sufficiently large, finite value for y is sufficient to obtain valid results. This will be discussed in more detail in chapter 6, where we present the results of the numerical work that was performed to solve the equations given in this chapter. 5.3 Summary Using an extension of the method of variable reflection and transmission amplitudes to the case of continuous unbound states, we have derived a set of differential equations which allow us to obtain the reflection and transmission 76 amplitudes. These equations do not require us to solve for the wave functions "4>ii.v{%), and as a result the calculations required to obtain the reflection and transmission amplitudes are considerably simplified. We have also obtained expressions for the reflection and transmission probabilities in terms of the intermediary quantities Ullv(y) and Qtw{y). 77 Chapter 6 Numerical Results for the Time-independent Case In this chapter, we use numerical methods to solve the equations for Riu,(y) and 7)j„(y) derived in the previous chapter using the method of variable reflection and transmission amplitudes. We consider the case of a molecule incident upon a delta potential barrier, subject to a double well binding potential with a single bound state and continuous unbound states. We first give the equations for eigenstates and eigenvalue conditions, then outline the numerical methods used to solve for R,iV{y) and TM,y(y), and finally describe and interpret our numerical results. 78 6.1 The Double Well Potential Recall that we considered a molecule with a delta well binding potential for the analytical work in chapter 4. While this potential well did capture important physical features of the molecule, it was chosen mainly for its relative analytical simplicity. For the numerical work the follows, we will make use of a more sophisticated binding potential: the double square well binding with a central hard core. In many earlier studies of molecular tunnelling [5, 6, 11, 12, 13], the molecular binding potential was approximated as follows: V2, 0 < |e| < a, Uo(Z) = { -V1} a<|e|<6, 0, (6-1) 6<|£|i n F n (a)cosh(s n O, 0 6, 80 (6.2) where F / ^ = cos[p»(6 - Q] + (rn/pn) sin[pn(6 - Q] cosh(s ?l a) g. and the terms pn, rn, and sn are defined by p^ = m 0 4 + Cn )/ft2 ) ( 6 .4) r2n = m.(-en)/tf, (6.5) s* = m(K2 - cn)/h2, (6.6) where en is the relative motion energy of the molecule in state n. The normalization constant An is quite complicated, and therefore we direct the interested reader to appendix A, where An and other normalization terms are defined explicitly. Note that because of the even symmetry of the eigenstates, it is not necessary to define (6.2) for negative values of £. The allowed bound state energy eigenvalues e„ are given by the equation sn ^ , , , pntan[pn(b - a) - rn — tanh(s n a = — - — j — ry. Pn r n t a n p n ( 6 - a)\+pn (6.7) By varying the parameters Vx, V2, a,, b, and m, we may create a molecule with an arbitrary number of bound eigenstates any value of energy. For this thesis, we vary the parameters so that the molecule has a single bound state. 81 For unbound states, the eigenstates are given by AqFq(a) cosh(.s?^), 0 < £ < a, x,(0 = A, cosh(s (/ a)F r/ (£), ^sin(rc/0, a < £ < 6, (6.8) e>^ where *UO = cos[p,(b - 0 ] sin[?-,b] 00811(550) r, sin[p,(b - 0 ] cosfob] pqcosh(sga) (6.9) eq)/h2, (6.10) and pq, rq, and sq are defined by p2q = m(Vl + rl = m(eq)/Ti2 (6.11) s] = m (V 2 - ej/ft 2 (6.12) where eq is the relative motion energy of a molecule in state q. Since the molecule has a continuum of unbound states, there is no need for an eigenvalue condition for the unbound states, i.e. all unbound energies eq > 0 are allowed. 82 6.1.2 T h e External Barrier; Dimensionless Quantities For the external potential, we choose a Dirac delta-barrier of strength A, given by V(xj) = X5(xj), (6.13) where j = 1, 2 and Xj refers to the coordinate of the j t h atom in the molecule. It is convenient at this point to introduce definitions of the dimensionless quantities we will be using in our numerical work. These definitions are as follows: x x = —, k = ka, a r - e" f* = Vi, A= M V* N = - , A , Zi&v = a Z^vi aVi lmVi 9 = a]j-w. (a -\A\ (6.14) The numerical values of the dimensionless parameters are chosen to be: g = 15, iV = 5, A = 0.01. It must be noted that these chosen values of the dimensionless constants are not arbitrary, but rather have been chosen so as to be comparable to realistic systems. For example, taking m, Vi, and a to be on the order of an atomic mass unit, an electron volt, and the Bohr radius respectively, one finds that g is of the same order as the value given above. The binding energies were determined by our choice of the parameter b/a. It is found that lower binding energies occur for smaller values of b/a, i.e. narrowing the wells results in a more weakly bound molecule. In this thesis, we consider two molecular states, differing from one other by the value of b/a. 83 For one molecular state, we choose b/a = 1.08, resulting in a dimensionless binding energy value of f0 = —0.0127; for the other, we choose b/a = 1.075, giving dimensionless binding energy /o = —0.0022. 6.1.3 Effective Potentials for a Molecule Incident Upon a Delta Barrier 50 - —-f„ = -0.0022 f„ =-0.0127 40 • 30 - 20 - 10 - \ J I 0- -10 -5 10 Figure 6.2: Plot of the effective potentials Z00(x) for two binding energies. The binding energies are indicated in the legend. We note that in this plot and the ones to follow, we omit the use of tildes to denote dimensionless quantities. Unless otherwise indicated, all graphs are plotted in dimensionless units. 84 As we showed in Chapter 4, the choice of an external barrier of the form (G.13) leads to effective potentials of the form Z»(x) = ^Xl(2x)x*(2x) Since we are able to express exactly the relative motion eigenstates (6.15) Xv(^x) and x(2x), we are therefore also able to use exact values for the effective potentials. In figure 6.2, we present plots of the effective potentials ZQQ(X) 0.8 0.6 -0.0127->3.737e-9 -0.0127->4.002e-6 -0.0022 - > 3.737e-9 -0.0022 --> 4.002e-6 0.4 A N 0.2 0.0 A Figure 6.3: Plot of the effective potentials Z0q{x). The transitions in relative motion energy to which these potentials correspond are indicated in the legend. 85 for f0 = -0.0127 and fQ = -0.0022. Figure 6.3 displays plots of the effective potentials Z0q(x), which correspond to transitions from the bound state to an unbound state q. The effective potential Z^v{x) may be interpreted as the potential barrier encountered by the CM of the molecule when incident in the state /J, and reflected or transmitted in state v. The extended nature of the molecule results in the single delta barrier being transformed into a pair of extended barriers. The smoothness of the barriers results from integrating over the relative motion coordinate £. Comparing figure 6.2 and figure 6.3, we notice that the effective potential Zoo(^) strongly dominates over the effective potentials Z0q(x). As well, examining figure 6.3, two trends may be observed. First, all other factors being the same, the peaks of effective potentials Zoq(x) for the stronger binding energy tend to be slightly higher than those for the stronger binding energy. Second, the effective potentials corresponding to a transition to a higher energy unbound state are much higher than those corresponding to a transition to a lower energy unbound state. Both of these results may be understood in terms of energy conservation. Greater transitions in molecular energy require a greater loss in the kinetic energy of the molecule. A greater loss in the kinetic energy, in turn, decreases the probability of transmission past the barrier. The loss in kinetic energy that a molecule undergoes when transitioning to a higher energy state is represented by the effective potential ZQq(x) as greater magnitude in the effective barrier. We return to the effective potentials when we examine the probabilities 86 of reflection and transmission in section 6.3. We next discuss the so-called "critical" wave number, which determines whether an incident molecule will have enough energy to transition to the continuum of unbound states. 6.1.4 The Critical Wave N u m b e r Recall the definition of the CM wave number kv given in chapter 2 for a molecule with energy E in the state v. kl = -^[E-eu\. (6.16) In order for a molecule incident in state \x to transition to a higher energy state v, the total energy E of the molecule must be high enough such that kl > 0, i.e. we must have E > e„. (6.17) The total energy E is determined both by the incident relative motion energy eM and the incident CM wave numberfcA,..If we rewrite E in terms of fcM and eM, substitute into equation (6.17), and solve for k^, we obtain ^,>lte(e,-e / ( ). 87 (6.18) The minimum value of kfJ at which a transition from state \x to state v is possible is referred to as the critical wave number kc , and is given by = /4m **" v n r(ev " e " ) - (6,19) Of particular importance is the critical wave number needed to transition from the bound state to an unbound state. The bound state energy is e0, and the lowest unbound energy is eq ~ 0. Thus, the critical wave number for transitions from the bound state to an unbound state is K=f^(-to)- (6.20) Note that the binding energy eo < 0, so fc£ is real. Rewriting in terms of the dimensionless quantities defined earlier, we get K =\/V(/,-U (6.21) K = ^V(-/0). (6.22) and For the molecule with binding energy f0 = —0.0127, the critical wave number for transitions to an unbound state is A;£ = 3.38; for the molecule with / 0 = —0.0022, ££ = 1.41. As we will show later in this chapter, the probabilities of reflection and transmission for a molecule incident with a 88 wave number below the critical value differ quite strongly from those for a molecule incident with a wave number above the critical value. We next briefly describe the numerical methods used to obtain the reflection and transmission probabilities. 6.2 Numerical Method In chapter 5, we presented a derivation of a set of differential equations, (5.41) and (5.42). which when solved allow us to obtain values for the reflection and transmission probabilities for a molecule incident upon a potential barrier. These equations greatly reduce the amount of calculation needed to obtain the probabilities. Nonetheless, a computer program is needed to perform the necessary numerical work. Indeed, the amount of numerical work needed to obtain the results in this thesis requires the use of a parallel-processing supercomputer. The results in this thesis were accomplished on the order of weeks using a parallel-processing computer. To obtain the same results using a 1 GHz processor would require a time of the order of years. The numerical method employed in this thesis was originally developed by Jeff Hnybida, and was earlier used to obtain the results for continuous unbound states given in reference [11]. The equations (5.41) and (5.42) are first order, ordinary, non-linear, coupled differential equations, and the boundary conditions for U^w{y) and Q,j.,y(y) as y —> co are known exactly. Thus, we employ a second-order back- 89 ward Runge-Kutta method to solve (5.41) and (5.42) for Ulw(y) and Qlw{y). The positive and negative infinity limits are replaced by finite values at which the effective potentials ZQv(y) are exponentially decreasing and very close to zero. For the results in this thesis, the limits are chosen to be y = ±10. In previous work, a step size of Ay = 0.001, or 1000 slices per unit y, was found to produce excellent convergence, in numerical results. We follow the same convention in this work, using 20000 slices to perform the Runge-Kutta integration. We use Simpson's rule to approximate the integral terms in equations (5.41) and (5.42) as a discrete sum over a set of equally spaced test points q. For this work, we use thirty-nine test points. The upper limits of the integrals are determined by the critical wave numbers corresponding to the test points. As a result of this, the spacing in the test points used for the numerical calculation varies as k increases, ranging from Aq — 0.01 for low k to A<7 = 0.05 for high k. We note that varying Aq values were also employed to obtain the results in [11]. Having outlined the numerical method, we proceed to the results of our calculation. 6.3 Results of the Numerical Calculation In this section we present a series of graphs showing the probabilities of reflection and transmission for a molecule incident in the bound state as a function of the dimensionless wave number k. All probabilities are plotted 90 1.2 1.0 0.8 J? 0.6 !Q ra -Q 8 a- 0.4 - 0.2 - 0.0 - 0 1 2 3 4 5 6 k Figure 6.4: Plot of P^b curves with respect to k for two binding energies. The binding energies for each case are indicated in the legend. from k = 0 to k = 5. 6.3.1 Probability of Reflection in the Bound State We first present the plot for the probability of reflection in the bound state, given in figure 6.4. We denote the probability of reflection in the bound state by P%b. The superscript indicates the molecule is incident in the bound state. The two curves denote the probabilities corresponding to molecules with binding energy / 0 = -0.0127 and / 0 = -0.0022. As mentioned in 91 section 6.1.4, the critical wave number for the molecule with binding energy /o = —0.0127 is kfu = 3.38, and the critical wave number for the molecule with binding energy f0 = —0.0022 is k^ = 1.41. For both curves there is a steep decline in the probability of reflection in the bound state at the critical wave number. The decline in P^ 6 is steeper for the more weakly bound molecule. This decline is due to the molecule being able to access a greater number of states upon gaining sufficient energy to reach the continuum. The curve corresponding to /o = —0.0022 shows an oscillation in the probabilities with respect to k, with peaks in the probability of varying height and width. The highest peak occurs at k = 3.2, and has a magnitude of Pfib = 0.23. A similar pattern of oscillation occurs in the /o = —0.0127 curve as well. The probability appears to peak with P^b > 0.4 for k > 5. The peak values of P%b for each of these curves suggests that a more weakly bound molecule, having sufficient kinetic energy to access unbound states, is less likely to be reflected in the bound state. Such a result is suggested by physical intuition. A more weakly bound molecule is less stable, and thus is more likely to break up upon interacting with an external barrier. Two pronounced minima in Pfib for /o = —0.0127 occur at k = 2.27 and k = 4.26. These minima, unlike those that occur in the / 0 = —0.0022 curve, indicate that the probability of reflection goes to zero for these values of k. The minimum at k = 2.27 occurs at a wave number less than the critical wa,ve number kcu for this molecule. Minima of this sort were observed 92 in previous work by Goodvin and Shegelski [5]. Recall figure 6.2, which plots the effective potential Z00(x) for both binding energies. The drop in bound state reflection probability at k = 2.27 occurs as a result of destructive interference between I/JOO{X) waves reflected from the first peak in the effective potential and waves reflected from the second peak. Similar phenomena have been observed in Fabry-Parot interferometers and in antirefiection films in optics. The minimum at k = 4.26 is also due to the same phenomenon. We note, as well, that a minimum in P^b occurs at k = 4.24 for the / 0 = —0.0022 curve. 6.3.2 Probability of Transmission in the Bound State Next, we consider P®b, the probability of transmission in the bound state, given in figure 6.5. For both binding energies, there is a peak in the probabilities, followed by a decrease in probabilities, for k > k^. The peak in the /o = —0.0127 curve occurs at k = 4.64 at magnitude Pj-b = 0.446, while the peak in the / 0 = —0.0022 curve occurs at k — 2.00 at magnitude P-% — 0.354. The slightly greater peak in the ,/o = —0.0127 curve suggests that the probability of transmission in the bound state, in general, decreases with decreased binding strength, as does the probability of reflection in the bound state. Interestingly, the probability of transmission in the bound state, for /o = —0.0022, seems to decrease almost linearly from k = 2.2 to k = 3.8, approaching nearly zero at k = 3.8. This minimum in P®b occurs fairly close in k to the minimum in P%b for the same binding energy. The minimum at 93 1.2 1.0 f0 = -0.0022 0.8 ^ 0.6 o £ 0.4 0.2 0.0 2 3 k Figure 6.5: Plot of P°b curves with respect to k for two binding energies. k — 3.8 is then followed by a linear increase in the probability, increasing at nearly the same rate that it was decreasing earlier. This sort of quasi-linear decrease and increase in probability has not been observed in our earlier work. For clarity, we note that the resonance in P^b at k = 2.27 for the f0 = —0.0127 curve was discussed in section 6.3.1. 94 0.5 f0 =-0.0127 f0 = -0.0022 !5 £ 0.2 o CL 0.1 - 0 1 2 3 4 5 6 k Figure 6.6: Plot of P^u curves with respect to k for two binding energies. 6.3.3 Probability of Reflection in an Unbound State It would seem, from the plots of P%b and P^b, that the probabilities of reflection and transmission in the bound state decrease as the binding energy decreases. When we consider the probabilities of reflection and transmission in the unbound state, we notice the opposite trend. In figure 6.6, we plot the probability of reflection in an unbound state, PR U , for both molecular states. The curve for ,/o = -0.0022 peaks at P£ u = 0.45, noticeably higher than the peak of P$„ = 0.35 for the / 0 = -0.0127 95 curve. In addition, while P^ u for f0 = —0.0022 does decrease for k « 2.4, it does not approach zero, as we would expect for increasing k (in general, the total probability of reflection goes to zero as the energy of the molecule increases). Indeed, P%, seems to roughly level off at P%u « 0.17 for k > 4. We note, however, that this trend could change for higher k. In any event, it would seem that the probability of reflection in an unbound state increases as the molecule becomes more weakly bound. We note that the apparent jaggedness of the P^u curve at k ~ 2.4 for /o = —0.0022 is not physical, but is rather a numerical artifact, resulting from the numerical approximation of integral terms by a discrete sum (i.e. Simpson's rule). 6.3.4 Probability of Transmission in an Unbound State The curves corresponding to transmission in an unbound state (P$„) are shown in figure 6.7. The curve for f0 = —0.0022 is particularly striking: an initial strong peak at k = 1.7, followed by an even stronger pseudoresonance at k ~ 4.3, where P$u = 0.72. Clearly, these curves suggest that the probability of transmission in an unbound state strongly increases with decreased binding energy. From the effective potentials plotted in figure 6.3, we expect this to be the case. The peaks in the effective potential Zoq(x) for a more deeply bound state are higher than those for a weakly bound state. Thus, as the kinetic energy increases, the probability of tunnelling in an unbound state shows a greater increase for a more weakly bound molecule. 96 0.8 f0 =-0.0127 f„ = -0.0022 / \ 0.6 \ £,0.4- \ \ \ 15 t I \ ( \/ o I 0.2 - 0.0 - 0 1 2 3 4 5 6 k Figure 6.7: Plot of P^u curves with respect to k for two binding energies. 6.3.5 Total Probability of Transmission Figure 6.8 displays plots the total probability of transmission, P$, for both of the molecular states observed thus far, as well as for a third state of binding energy /o = —0.0296. Apart from the resonance near k = 2.27, the probability of transmission for the / 0 = -0.0022 molecular state is higher than that of the / 0 = -0.0127 molecular state for all k. Again, this is to be expected from the plots of the effective potentials in figures 6.2 and 6.3. The effective potentials Z00(x) and Z0q(x) for the weakly bound molecule are of 97 1.2 f0 --0.0127 1.0 f0 =-0.0022 f0 =-0.0296 0.8 / | T 0.6 / !Q TO \ ^—7 .a o £ 0.4 0.2 0.0 / \ \ \ \ / \ II Figure 6.8: Plot of B® curves with respect to k for three binding energies. We have added the third graph, for energy ,/0 = —0.0296, to determine if the correlation observed at k = 2.27 occurs at higher k values. lower height than those for the strongly bound molecule, and thus we should expect greater transmission for a weakly bound molecule. We note at last an interesting correlation between the Pj> curves near the resonance at k = 2.27. There appears to be a correlation between the resonance in Pj. for ,/0 = —0.0127 and a local minimum in B® for /o = —0.0022. It is this observation which prompts us to plot these two curves along with the Pj- curve for a molecule with binding energy /o = —0.0296 in order to determine if similar trends are observed for higher k. Two resonances are observed for f0 = -0.0296, one at k = 2.41 and the other at k = 4.90. The critical wave number for this more strongly bound molecule is k = 5.16. Thus, for all k shown on the graph, the molecule with binding energy /o = —0.0296 is unable to access unbound states, and the resonances observed are therefore a result of the same phenomenon that creates a resonance at k = 2.27 for the /o = —0.0127 curve. However, no correlation is observed between the resonance at k = 4.90 for f0 = —0.0296 and the P? curves for the other two molecular states. This suggests that the correlation that occurs at k = 2.27 may be coincidental, or may by a phenomenon unique to low k. 6.3.6 Comparison of Numerical Results to the Analytical Predictions The numerical results shown in sections 6.3.1 to 6.3.5 would seem to oppose the prediction made in chapter 4, i.e. that the probability of transmission in the bound state approaches unity in the limit of arbitrarily weak binding, while all other probabilities approach zero in the same limit. However, there are two facts to bear in mind. First, while we have investigated the decreasing binding strength, we have in no way approached the limit of arbitrarily weak binding. Recalling figure 6.2, the effective potentials Z00(x) for both molecular states have not vanished. The vanishing of the effective potentials for the delta potential well in the limit of arbitrarily weak binding is what 99 led to the results predicted in chapter 4. Second, the numerical results given in this chapter assume a double well potential. The double well potential is far more complicated than the delta well, and thus is likely to behave in a more complicated way than the delta well as the binding energy approaches zero. 6.4 Summary We have used numerical methods to solve the differential equations derived in chapter 5. From the results of our numerical analysis, we have presented plots of the probabilities of reflection and transmission in the bound state and in an unbound state for two molecular states of different binding energy. We have found, in general, that the probabilities of reflection and transmission in the bound state decrease as the molecule becomes more weakly bound, while the probabilities of reflection and transmission in an unbound state increase with weaker binding. We have also found that the total probability of transmission increases with weaker binding. An interesting possible correlation was found between the transmission resonance of the more strongly bound molecule and a local minimum in the total probability of transmission for the more weakly bound molecular state. 100 Chapter 7 Time-dependent Tunnelling In this chapter, we use Crank-Nicholson integration to numerically model the tunnelling of a molecule with discrete unbound states across a potential barrier. This analysis differs from that of the previous chapters in that it is timedependent, while the earlier results were obtained using time-independent analysis. Our results allow us to give a qualitative description of the tunnelling of a molecular wave packet. We find that the molecule has a high likelihood of straddling the barrier. This outcome is not considered in timeindependent analyses of molecular tunnelling. We also determine the probabilities of reflection and transmission as functions of the CM wave number by making use of the long-time probabilities. 101 7.1 The Double Well Potential with Finite Interaction Range Recall, from chapter 2, that we defined the wave function \P(x,£,£) of a molecule incident upon a potential barrier in time as (7.1) *(x,£,t) = X>0(O^M)> where the functions x(0 a r e obtained by solving the relative motion Schrodinger equation V M ,,(x, t)~2^ Z^(x)^{x, 4> t) = - 1 7 — ' — dt , (7.3) where 4m, r°° Zv4>(x) = - ^ / d^ x%{Qx»{C "M^H). iin 4m 7 = 102 . (7-4) (7.5) (7.6) and e„ is the relative motion energy eigenvalue for state v. As always, we use Greek letters to denote both bound and unbound states. For the binding potential, we choose a double square well with finite interaction range L: Uo(0 = v2, 0 < |£| < a, -Vu a < |£| < b, o, b<\^\ L . (7.7) Solving (7.2) for the potential given in (7.7), we obtain exact expressions for the (even) relative motion states Xn(0 an d X?(0 : AnFn(a) cosh(s„£), 0 < £ < a, Xn(0 = An cosh(.s„a)F7,,(0, Ansmh[rn(L - 0], 0, a < £ < b, (7b < £ < L, e > L, and x,(0 >l,F,(a)cosh(s,0, 0 L, 103 (7.9) where F F (e) = cos[p„(6 - £)] sinh[r n (L - b)\ cosh(s n a) rH sin[pn(fr - Q] cosh[r n (L - b)} pn cosh(s„a) (7.10) ,c\ = c o s b g ( & - Q ] s i n [ r g ( L - b ) ] 9 cosh(s g a) | rgsin[p,(6-Q]cos[r,(L-&)] p , cosh(s a and the quantities p„., r„, s„, p, /; rf/, and sq are defined as they were in chapter 6. The normalization constants An and Aq, like those in chapter 6, are given in appendix A. The eigenvalue conditions for the bound and unbound states are given as follows: sn . , , s Pn tan[p„(6 - a)} tanh[r n (L - b)} - rn — tanh(s n a) = —.—— --. —,—rr, Pn Pn tanh r„,(L - b)\ + rn tan[pn{b - a)\ sq Pq .( (i-12) s _ pq tan[pq(& - a)) tan[r (/ (L - b)} - rq Pq t a n ^ L - b)\ + rq tan[pq{b - a)J For the external barrier we once again choose the Dirac delta barrier, V(xj) = X5(Xj), (7.14) where j = 1, 2 and x0 refers to the coordinate of the j t h atom in the molecule. In time-dependent studies of single particles modelled as wave packets, sharp 104 potentials like the delta barrier are typically avoided, and instead smoother barriers, like the Gaussian barrier, are employed. However, because of the "smoothing" of the effective potential, we may examine the time-dependent tunnelling of a molecule incident upon a delta barrier without difficulty. Figure 7.1: Plot of Z00(x), Z 01 (x), and Zn(x). These effective potentials correspond to molecular energies /o = —0.0001186 and fi = 0.001046. For the numerical work conducted in this chapter, we make use of the dimensionless quantities defined in (6.14). In addition, we define the dimen- 105 sionless quantities t and 7, t t = ~, Ama 1=^ ~ , r 7.15 TIT where r is a time constant determined by our choice of 7. For reasons of numerical convenience, we choose 7 = 10. In addition, we choose g = 15, N = 5, A = 0.01, as in chapter 6. The interaction range is represented by the dimensionless parameter L/a, which we choose to be equal to 10 (this value for the interaction range has been used in previous work [11]). As in chapter 6, we use the parameter b/a to determine the binding energy. In this work, we investigate a molecule with a single bound state and three discrete unbound states. Choosing b/a = 1.0775, we have f0 = —0.0001186, /1 = 0.001046, f2 = 0.003223 and f3 = 0.006489. The dimensionless effective potentials ZQ0(x), Zoi(x), and Zn(x) are plotted in figure 7.1. We next discuss the numerical method used to solve equation (7.3). 7.2 Numerical Method The method used to simulate the time-evolution of the molecule as described by equation (7.3) is briefly outlined. Our method is based on the well-known Crank-Nicholson method for numerically solving heat equation problems [18]. The x-derivative is discretized using a centred finite difference method, with Ax = 0.01. Dirichlet boundary conditions are imposed at 106 x = ±75 (these boundary conditions effectively define the range of x-values). The time-integration is performed using the Crank-Nicholson method, with At = 0.001. We use Crank-Nicholson integration because the method is unconditionally stable, and because convergence is second order in time, i.e. error is proportional to At2. Conversely, convergence of Euler methods is only first order in time (i.e. oc At1), and such methods are not unconditionally stable. For our analysis we must consider the effects of coupling between different molecular states. As part of the numerical integration process, the secondorder .x-derivative and the coupling of states via the effective potentials is represented by a square matrix, while the function being solved for is represented by a discretized column vector. If the square matrix is symmetric and positive definite, then there is guaranteed to exist an LU-decomposition for the matrix. This, in turn, means that the matrix may be easily inverted and the time integration thus may be easily performed. Our method depends on the matrix having an LU-decomposition. However, because of the coupling that occurs between states via the effective potentials, the square matrix is not symmetric, and thus an LU-decomposition is not guaranteed to exist. One of the reasons that our numerical work was limited to a spatial span from x = —75 to x = 75, and to a molecule with only four states, was because LU-decompositions failed for greater .x-spans and higher numbers of states. However, it should be noted that the failure of some of these decomposi107 tions was not necessarily due to the non-existence of the LU-decomposition, but rather because the decomposition required a greater amount of computer memory than was available to us. To perform LU-decomposition, as well as the Crank-Nicholson time integration, we required the use of a computer cluster. Unlike with the time-independent results, for which processing time was the deciding factor in employing a supercomputer, for the time-dependent case it was a lack of sufficient computer memory which drove us to make use of a supercomputer. To perform a single run of the simulation, which involved integrating over a span of 50,000 time spaces, we required 20 Gb of memory. We next discuss the results of the numerical simulation. 7.3 Results of t h e Numerical Analysis We consider a molecule incident upon the potential barrier in the bound state. The functions ij)v{x,t) in equations (7.1) and (7.3) correspond to the CM motion of a molecule in state v. This means that if, at time t, the molecule occupies only one state v', then the wave functions for all ip,^,/{x, i) are zero for all x. Thus, for a molecule incident in the bound state, we set 1/^0(2, 0) = 0. Since the initial state of the molecule is chosen in this way, we omit the subscript // used in our time-independent work to denote the initial state. We model the initial molecular wave function 4>o(x, 0) as a normalized Gaussian wave packet. Expressed in terms of dimensionless arguments, we 108 have MW) = ( , 1 ) efe-^1, (7.16) where XQ is the initial expectation CM position of the wave packet, k is the expectation CM wave number, and a is the standard deviation (the "width," so to speak) of the wave packet. Recall that for a wave packet, the Heisenberg uncertainty principle dictates that both the CM position and CM wave number cannot both be known with exact precision. Hence, the wave packet is spread out in both x and k. This is why we use the terms expectation position and expectation wave number, as opposed to simply position and wave number. For this work, we set x0 — —15 and a = 1.5, and examine the tunnelling of the molecule for expectation wave numbers k = 1.00,1.25,1.50,1.75,2.00,2.25, and 2.50. We chose the upper limit of k = 2.50 so as to avoid approaching the critical wave number for higher energy eigenstates which are not included in this simulation. We note that the critical wave number for transition from the bound state to the lowest energy unbound state is fcoi = 1-024, just above our lowest expectation wave number k = 1.00. Before moving on to our results, we first define the quantities PR„(£) and P'fu(t), which we refer to, loosely, as the probabilities of "reflection" and "transmission" in state v at time t. We say loosely because these quantities only tell us the likelihood of observing the molecule in state v behind or ahead of the barrier at some time I. To obtain the true probability of 109 reflection and transmission requires evaluating Pn„(t) and Prv(t) for large t. We also define a new quantity, lDsv{t), the probability of "straddling," i.e. of observing the molecule within a close vicinity of the barrier at time t. For all future references to these quantities, we dispense with quotation marks. The quantities are defined, in terms of dimensionless quantities, as follows: \t/;„(x,t)\2dx, PRu(i)=f\ fxrn.ax ^ (7.17) ^ |^(.r,t)| 2 rf.r, (7.18) \MxMdx- (7.19) PTu{t)= Jo Ps,{i)= I J-5 The quantities ±.xmax denote the upper and lower spatial limits of the numerical simulation, i.e. the location in space where Dirichlet boundary conditions are imposed. For this work, xmax = 75. The limits of integration for Ps,;(t) correspond to the range of the effective potential in x (for L/a = 10, the effective potential vanishes for \x\ > 5). From (7.17), (7.18), and (7.19), we may define the probabilities of reflection, transmission, and straddling in the bound state and in an unbound state as follows: Pm(i) = Pm{i), PRu(t) = J2 PRS)^ (7-20) Prb{i) = Proii), Pm(i) = £ (7.21) <^0 110 /%(*), Psb® = Pso®, Psu(i) = £ Ps^t)- (7.22) Finally, we define the probability of finding the molecule in an unbound state at time t: Pb^u{i) = PRu(J) + PTu{i). (7.23) Having defined the above quantities, we next present our analysis of the time-dependent tunnelling of a molecule. 7.3.1 Reflection in the Bound State and in an Unbound State vs. k In figure 7.2, we present plots of Pn,b(t) for all seven k values as a function of dimensionless time. In all curves, we notice steep drops in PR(>(£) for short to medium-range times followed by gradual increase in probability. As we will explain later, this drop in probability is due to the molecule transitioning to an unbound state upon contact with the barrier. For greater values of k, these drops in probability are more pronounced. For long-range times (i > 100), the probabilities tend to level off, displaying slight oscillation with time. The peak to peak magnitude of these long-time oscillations is never greater than 0.02 (note that the range of probabilities in figure 7.2 is 0.85 to 1.00). Thus, the t = 200 values serve as a fairly good approximation to the true probabilities of reflection in the bound state. These curves may be interpreted as follows. The molecule, incident in the 111 1.02 1.00 0.98 0.96 !;u\ \ l»\\ i in _>* = 111 > \ 'i|!\ \ 0.94 I! • \ tc D- IiI \ .a 2 0.92 0.90 0.88 0.86 0.84 v / 111 1 \ |M I. \ / /%/ 11 t / k= 1 II\ \ -.>' k = 1.25 11 1 ^ '• \\ \/ k= 1.5 k = 1.75 k=2 / \\ k = 2.25 v>^ k = 2.5 /*?Vv. Vv 50 100 150 200 250 t Figure 7.2: Plots of PRb{i) for all k. Once again, we note that all graphs are plotted in terms of dimensionless quantities, and that the tilde superscripts have been omitted. bound state upon the potential barrier, undergoes a temporary transition to a higher energy state, followed by transition back down to the bound state and reflection away from the barrier. The transition to a higher energy state occurs behind the barrier (i.e. x < 0), without tunnelling occurring. This process may be likened to an elastic ball hitting a hard wall: the ball hits the wall, compresses, and then bounces back from the wall. A plot of (1 - PRb(t)), PTb{t), PRuit), and PTu(l) for k = 1.00, given in figure 7.3, 112 0.030 0.025 0.020 - 1-Pp PT £ 0.015 CO .a £ 0.010 0.005 0.000 H 50 100 150 200 250 t Figure 7.3: Plots of (1 - PRb{t)), PTb(i), PRu(i), and PTu(t) for k = 1.00. supports this interpretation. For short to medium times (i < 75), there is a strong correlation between the curves corresponding to (1 — PRb{t)) and Pnu(t)- No such correlation exists between (1 — Pnb{i)) and the other curves. Similar trends are observed in plots of these quantities for all other k values. Referring again to figure 7.3, we noted earlier that the magnitude of the drops in PRb{i) increases with higher k. We expect for this to occur: a molecule with higher kinetic energy is more likely to transition to an unbound state than one with low kinetic energy. 113 We note, finally, that the transition to an unbound state occurs for k = 1.00, a wave number lower than the critical wave number. The transition is thus, from a classical perspective, forbidden. It is, however, allowed under the Heisenberg uncertainty principle for time and energy, which allows for temporary transitions to classically forbidden energies. 7.3.2 Analysis of Molecular Straddling Recall figure 7.3. For long times [l > 100) there appears to be a correlation between PRU{J) and Pruit)- Specifically, the two curves appear to oscillate about some average value. As well, the maxima of each curve seem to correspond to the minima of the other curve, and vice versa. These curves suggest that a portion of the wave packet oscillates about the barrier. That is, a molecule that transitions to an unbound state, if it does not reflect from the barrier in the way described in the previous section, will straddle the barrier. This interpretation is supported by figure 7.4, which plots the difference between the probability of transition from the bound state to an unbound state, Pb-^u(t), and the probability of straddling in an unbound state, Psu(t)For all k, the difference between the two probabilities approaches a constant value following an initial increase at short times. The largest difference, corresponding to k — 2.5, is less than 0.03. For the lowest k values, the difference is less than 0.005. In figure [?], we plot (Pb^u(t) - Ps„.(i)) for all k. These plots show that the probability of straddling in an unbound state 114 u.uou k=1 k = 1.25 k = 1.5 k = 1.75 k=2 k = 2.25 k = 2.5 0.025 - -*- y~~ / / 1 o o en 1 1 / 1 1 - ' " ^ p o o Probability 0.020 - / y y / ^ • • - " " __—--• p o a\ / / i i //,-~ ..-•- ^±^- 0.000 - 1 i i i i 50 100 150 200 250 t Figure 7.4: Plots of (P 6 _ u (£) - PSu(i)) for all fc. does not vanish for long times. In figure 7.5, we plot PSu(i), Pb^u{t), (1 - Pm(t)) and PTb(t) for fc = 2.5. Notice that even when the difference between Psu(t) and Pb^u(t) is relatively large for this value of A;, there is nonetheless a noticeable correlation between the two curves. As well, there is a strong anti-correlation between Psu{t) and the probabilities Pn,b(t) and Prb(t)- This indicates that the molecule is oscillating between the bound state and unbound states at long times. The anti-correlation is more noticeable and consistent in the PrbiJ) curve. 115 0.16 0.14 1-PD PT 0.12 0.10 = 0.08 ' V \ \ /'\ \ A \ / \ r~\ S 0.06 N.-- \ 0.04 0.02 -\ 0.00 50 100 150 200 250 t Figure 7.5: Plots of PSu(i), Pb^u(t), (1 - PRb{t)) and PTb{t) for all k. The physical interpretation of this is that the molecule, in approaching the barrier, has some probability of transitioning to the unbound state near the barrier, and then tunnelling past the barrier in the bound state. Multiple oscillations between the bound and the unbound state may occur before the molecule eventually tunnels past the barrier in the bound state. The results for the unbound probability of straddling suggest a physical outcome for the molecule which we never considered in time-dependent investigations. Namely, there is a certain probability that a molecule, incident 116 upon a potential barrier, breaks up in such a way that the atoms travel in opposite directions with equal magnitudes of momentum. The atoms do not come to a standstill, since they transition to a higher energy relative motion state. Another possibility suggested by these results is that the atoms oscillate at a high rate at some average distance away from one another, i.e. they don't fly apart (recall the molecule has finite interaction range). The atoms cannot travel with different momenta since in this case the CM would not straddle the barrier. Earlier results, which used time-independent analysis, assumed that the CM must either reflect from or tunnel past the barrier. Straddling of this sort, to the best of our knowledge, is a new discovery, which wouldn't have been made using only time-independent methods. Finally, we present in figure 7.6 the probability of straddling in the bound state. These results show that the molecule does not straddle in the bound state for long times. That is, it either reflects from the barrier or tunnels past it. The strong peaks at early times correspond to the initial interaction of the wave packet and the barrier. The decline at medium-range times corresponds to the wave packet reflecting from or tunnelling past the barrier. The relatively slow decline for the k = 1.00 curve is simply a consequence of the low speed of the wave packet. The oscillations observed for the curves at long times are most likely a consequence of the oscillation between bound and unbound states that occurs for a molecule tunnelling in a bound state. 117 250 Figure 7.6: Plots of PSb{i) for k = 2.5. 7.3.3 Probabilities as Functions of k For all probabilities of reflection, transmission, and straddling, the long time trend is toward oscillation about some constant value. Recalling figure 7.2, the peak-to-peak magnitudes of the oscillations in the long time PRb(t) curves never exceeds 0.02. A similar trend is observed for Prb(i), f/?,«(£), and Pru(i) (we omit the corresponding graphs for presentation purposes). Thus, to a fair approximation, we may plot the "true" probabilities of reflection, transmission, and straddling as functions of k using the t = 200 values of Pnb(t), 118 Prb{i), and so forth. 0.14 -i 0.12 A 0.10 - 0.08 - i5 _§ 0.06 o CL 0.04 0.02 0.00 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 k Figure 7.7:_ (1 - Plib(i)), respect to k. PTb{i), PRu(t), PTu(i) and PSu(l) at t = 200 with In figure 7.7, we plot [1 - P Hb (f)], P T 6 (i), P flu (f), /V«(*) and PSu{t) at £ = 200 as funtions of fc. In all five curves, we observe non-linear trends in probability with respect to k. As well, there is a lack of resonant structure, i.e. the probabilities are neither zero nor one for any k. Clearly, the probability of reflection in the bound state dominates over all other probabilities. For increasing k, the probability of reflection in the bound state decreases. We expect this decline in reflection probability to occur, since 119 greater k corresponds to greater kinetic energy, which in turn corresponds to a higher likelihood of tunnelling in the bound state or transitioning to a higher energy state. The probability of transmission in the bound state reaches a local maximum of approximately 0.07 at k ~ 2.25. We comment on this below. As well, for k > 1.6, the probability of transmission in the bound state is greater than the probabilities of reflection, transmission, and straddling in an unbound state. We note that for k < 1.4 the probability of transmission in the bound state is, interestingly, less than the probabilities of reflection, transmission, and straddling in an unbound state. The probability of straddling in an unbound state has a local maximum of approximately 0.03 at k « 1.75. For most k in this range, the probability of straddling in an unbound state is greater than the probabilities of reflection and transmission in an unbound state. However, for k = 2.50, the probabilities of reflection and transmission in an unbound state exceed the probability of straddling in an unbound state. This means that as the kinetic energy of the molecule increases, the molecule is less likely to straddle in an unbound state. This could be understood, physically, in terms of energy considerations. A molecule which straddles has just enough kinetic energy to transition to an unbound energy. Because the CM kinetic energy is converted almost entirely to relative motion energy, the CM comes nearly to a standstill. A molecule with higher energy, however, will still have enough CM kinetic energy to continue past the barrier or reflect away from it in an unbound state. Finally, we note that the decrease in the probability of tunnelling in the bound state is 120 correlated to the increases in the probabilities of reflection and transmission in an unbound state in the region k > 2.2. 7.4 Summary In this chapter, using a time-dependent formulation, we investigated the tunnelling of a molecule incident in the bound state upon a delta potential barrier. Using plots of the probabilities of reflection, transmission, and straddling, we were able to gain a qualitative understanding of the mechanics of molecular tunnelling. We found that if a molecule is reflected from the barrier in the bound state, it is likely to temporarily transition to an unbound state before reflecting in the bound state. We also found that a molecule which transitions to an unbound state is more likely to straddle the barrier than it is to be reflected or to tunnel past the barrier. At the largest value of k, the probabilities of reflection and transmission in an unbound state exceed the probability of straddling in an unbound state. Finally, we found that a molecule which tunnels past the barrier in the bound state will first temporarily oscillate between being in the bound state and being in an unbound state while it is near the barrier. 121 Chapter 8 Conclusion 8.1 Summary of this Work In this thesis, we presented a study of the tunnelling of a diatomic molecule with a single bound state incident upon a potential barrier. Both time- independent and time-dependent formulations were examined, and we obtained both analytical and numerical results. A summary of the main findings is given next. In chapter 2 of the thesis, the formulation of the problem was given. We wrote the molecular wave function in centre-of-mass (CM) coordinates and expanded the solution in terms of the relative motion eigenstates. Doing this allowed us to extract the relative motion and write the problem in terms of multi-channel Schrodinger equations. The solutions to the multi-channel Schrodinger equations were wave functions corresponding to a molecule inci- 122 dent in state /.<. and reflected or transmitted in state v. The multi-channel formulation allowed us to model the CM as a single particle tunnelling through an effective potential dependent on both the external potential barrier and the relative motion eigenstates. Multi-channel Schrodinger equations were derived for both the time-independent and the time-dependent formulations. In chapter 3, we gave the formal solution to the time-independent multichannel Schrodinger equations. From this formal solution, we were able to define the reflection and transmission coefficients for a molecule incident in state ji and reflected or transmitted in state v. We were also able to determine the corresponding probabilities of reflection and transmission by considering the fluxes of the incoming, reflected, and transmitted waves. In chapter 4, we considered the presumably simple case of a molecule with a delta well binding potential of arbitrarily weak binding strength. We chose this binding potential for its mathematical simplicity, as well as because, for any binding strength, there exists only a single bound state. First, we considered the case of such a molecule incident upon an infinite barrier in the bound state. While it was clear that the molecule would have to reflect, and while physical intuition suggested that the molecule would break up upon reflection, obtaining the exact values for the reflection coefficients proved difficult. Nonetheless, we were able to obtain important physical results. One of these results was that there exists a certain special distance from the barrier, xmax. beyond which the bound state ceases to exist. That is, for x > xmax, the bound state does not exist. We found that this distance becomes arbitrarily 123 large (i.e. xmax —> oo) as the molecule becomes arbitrarily weakly bound. Another result obtained was a set of asymptotic expressions for the molecular wave function in the limits x —* — oo and x —> xmax~. Investigation of this function, at the time of writing, is ongoing. In addition to the infinite barrier, we also examined the case of a molecule with a delta well binding potential incident upon a delta potential barrier. We found that the effective potentials corresponding to molecules incident in the bound state and reflected or transmitted in the bound state vanish as the molecule becomes more weakly bound. This result, in turn, suggested that the probability of transmission of a weakly bound molecule incident in the bound state approaches unity. This result was at odds with physical intuition, which suggested that an arbitrarily weakly bound molecule would break up upon contact with the barrier, resulting in the probabilities of reflection and transmission in the bound state approaching zero. In chapter 5, we employed Razavy's method of variable reflection and transmission amplitudes to obtain differential equations which, when solved, yield the reflection and transmission coefficients. These equations, unlike the expressions for the reflection and transmission coefficients obtained from the formal solutions, have no dependence on the CM wave functions Vv( x )> allowing for a great simplification in the calculations needed to obtain the reflection and transmission amplitudes. In chapter 6, we presented our numerical results for the time-independent formulation. We considered the case of a molecule with a double square well 124 binding potential with continuous unbound states. The potential was chosen so that molecule had a single bound state. We considered molecular states having two different binding energies, and calculated the probabilities of reflection and transmission in the bound state and in an unbound state as functions of the CM wave number. We found that the probabilities of reflection and transmission in the bound state tend to decrease with decreased binding energy, while the probabilities of reflection and transmission in the unbound states tend to increase with decreased binding energy. These results seemed to oppose the analytical result from chapter 4 stating that the probability of transmission in the bound state approaches unity for arbitrarily weak binding. We pointed out, however, that in chapter 6 we were examining a molecule with a double well potential, as opposed to a delta well potential, and that the binding energies, while small, did not approach the limit of arbitrarily weak binding. Although the results of chapter 4 and chapter 6 do not provide a clear conclusion, further work could do so. We had hoped the numerical work described in chapter 6 would resolve the issue. It is unlikely that numerical work could be expanded to the point where a definitive conclusion would be obtained. Instead, further analytical work may well provide a full resolution to the questions of probabilities of reflection and transmission in the bound and unbound states for an arbitrarily weakly bound molecule incident upon an external barrier. In chapter 7, we gave our results for the numerical calculation for the timedependent formulation. We once again examined a molecule with a double 125 well binding potential; in this case we assumed discrete unbound states. Unlike with the time-independent formulation, for which we employed various analytical methods to simplify the calculation, direct numerical integration was needed to solve the time-dependent multi-channel Schrodinger equation. We modelled the time-dependent CM motion as a Gaussian wave packet, and examined the tunnelling behaviour of the molecule for multiple CM wave numbers. Several qualitative observations were made concerning the process of molecular tunnelling. We found that reflection of the molecule in the bound state was likery to involve a temporary transition to an unbound state. We also found that a molecule which tunnels past the barrier in the bound state undergoes. many transitions between the bound state and unbound states as it passes the barrier. Most interestingly, we found that the molecule, when it transitions to an unbound state, has a high likelihood of straddling the barrier, i.e. neither reflecting nor tunnelling. This is an outcome that was not considered in time-independent formulations of molecular tunnelling. Straddling was not observed to occur in the bound state for long times. Using long time values for the probabilities, we constructed a plot of various probabilities as functions of the expectation value of the CM wave number. We found that the probability of reflection in the bound state dominated overall other probabilities, although it does decrease as the expectation value of the CM wave number increases. We also found that for a certain range of CM wave numbers, the probability of straddling in an unbound state is greater than the probabilities of reflection and transmission in an unbound 126 state. For higher wave numbers, however, the probabilities of reflection and transmission in an unbound state surpass the probability of straddling in the unbound state. 8.2 Future Work As mentioned earlier, investigation of the molecule incident on the infinite barrier is ongoing. These results of this study will appear in a future publication. As well, now that time-dependent modelling of molecular tunnelling has been demonstrated in this work, there are many possible avenues for further work in this area. For instance, our analysis has been proven viable for the case of a small number of discrete states. Therefore, an immediate application of methods developed for this thesis would be toward the tunnelling of a molecule with multiple bound states, without consideration of the unbound states. Time-independent work in this area has already been undertaken by Goodvin and Shegelski [5] for a small number of bound states, and by Hnybida and Shegelski for a large number of bound states [12]. Comparison between the results in these studies and those obtained using the time-dependent formulation will be of interest. Another area of interest would be an expansion of time-dependent investigation to three dimensions. In three-dimensional analysis the vibrational and rotational modes of the molecule are considered. 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