Solving the Time-Dependent Schr¨odingerEquationa David G. Robertsony Department of Physics, Otterbein University, Westerville, OH 43081 (Dated: October 10, 2011) Abstract Methods for solving the time-dependent Schr¨odingerequation in one dimension are discussed. Possible simulation projects include the study of scattering by barriers and wells, the analysis of time evolution by expansion in energy eigenstates, and tests of time-dependent perturbation theory. Keywords: quantum mechanics, Schr¨odingerequation, time dependence, scattering, perturbation theory, Crank-Nicholson method a Work supported by the National Science Foundation under grant CCLI DUE 0618252. y Email: [email protected] 1 CONTENTS I. Module Overview 2 II. What You Will Need 2 III. Quantum Mechanics Background 3 IV. Solving the Time-Dependent Schr¨odingerEquation 5 V. Simulation Projects 12 References 16 Glossary 18 I. MODULE OVERVIEW This module discusses methods for solving the time-dependent Schr¨odingerequation in one space dimension. This is the central problem in quantum mechanics, and courses in quantum physics typically devote considerable time to developing solutions in analytically tractable cases (e.g., square wells, the harmonic oscillator). The ability to determine the wavefunction numerically allows exploration of much wider class of potentials, however, as well as the study of e.g., scattering with actual wavepackets rather than plane waves. One can also test the predictions of time-dependent perturbation theory. Such exercises allow the development of deeper intuition regarding the properties of quantum systems. II. WHAT YOU WILL NEED The minimal physics background required includes quantum mechanics at the level of a typical sophomore-level course on modern physics, specifically the basic properties of wavefunctions and the Schr¨odingerequation. Students will ideally have worked through the free particle in one dimension and be familiar with wave packets, dispersion, and so on. An upper-level course in quantum mechanics will allow a richer exploration of problems with the tools developed here, for example the study of time-dependent perturbations. 2 On the mathematical side, facility with differential and integral calculus is essential, as is a basic familiarity with differential equations and complex numbers. Students who have passed through a sophomore-level modern physics course and the standard calculus sequence through the sophomore year should have the necessary background. The computing resources needed are actually fairly minimal, as the \size" of the compu- tational problem is modest. The natural framework for scientific computing is a high-level language like Fortran, C or C++, and this is an excellent project for students to sharpen their programming skills. However, the needed calculations can be done using Matlab or Mathematica. Support for complex arithmetic is essential. Some facility for generating plots will also be necessary, for example gnuplot or (less ideally) Excel. Some suggestions in this regard are given with the simulation projects. III. QUANTUM MECHANICS BACKGROUND The central problem of quantum mechanics is to solve the Schr¨odinger equation, which determines the time evolutions of the wavefunction for a system [1]. For a single non- relativistic particle in one dimension this takes the form @Ψ(x; t) h¯2 @2Ψ(x; t) ih¯ = − + V (x; t)Ψ(x; t); (3.1) @t 2m @x2 where m is the particle mass and V (x; t) is the potential energy. Given an initial wavefunc- tion Ψ(x; 0), the Schr¨odingerequation determines Ψ for all later (and earlier) times. The physical meaning of Ψ is that it gives the probability density for finding the particle at different locations, if its position is measured [3]. Specifically, the probability dP to find the particle in a small range from x to x + dx at time t is given by dP = jΨ(x; t)j2dx: (3.2) The probability to find the particle in some finite interval, say between x = a and x = b, is then the sum of the probabilities for each infinitesimal interval between a and b: Z b P = jΨ(x; t)j2dx: (3.3) a A critical requirement for the consistency of the framework is that the total probability to find the particle somewhere be unity; hence we must require that Z 1 jΨ(x; t)j2dx = 1: (3.4) −∞ 3 Wavefunctions that satisfy this condition are said to be \normalized." An important feature of the time evolution defind by the Schr¨odingerequation is that if the wavefunction is normalized at one time, then it will be normalized for all other times as well. This property is known as \unitarity." The physical importance of the normalization requirement cannot be over-emphasized; it is this condition that puts the \quantum" in quantum mechanics. Wavefunctions that do not satisfy this fall into two classes: functions for which the normalization integral is finite but not equal to one, and functions for which the normalization integral is infinite. Given a function in the first class, we can easily produce a normalized wavefunction. Assume that we have found a solution of the Schr¨odingerequation Ψ for which Z 1 jΨ(x; t)j2dx = A; (3.5) −∞ with A a finite number. Then the rescaled wavefunction 1 Ψ0(x; t) = p Ψ(x; t) (3.6) A will be properly normalized, i.e., will satisfy Z 1 jΨ0(x; t)j2dx = 1: (3.7) −∞ Note that because the Schr¨odingerequation is linear in the wavefunction, the rescaled wave- function remains a solution. Functions for which the normalization integral is infinite, on the other hand, cannot be normalized at all. Such wavefunctions are simply unacceptable for describing real physical systems, and must be discarded despite being solutions to the Schr¨odingerequation [4]. The problem of finding solutions to the Schr¨odingerequation is usually approached using the technique of separation of variables. This leads to the \time independent" Schr¨odinger equation, which determines the energy eigenstates (x): h¯2 d2 (x) − + V (x) (x) = E (x); (3.8) 2m dx2 Here E is a constant to be determined by solving the equation [5]. It has the form of an eigenvalue equation, H^ = E ; (3.9) 4 where H^ is the Hamiltonian operator h¯2 d2 H^ = − + V (x) (3.10) 2m dx2 and E is the eigenvalue. It should be emphasized that the TISE determines both and E; that is, we find that each solution works only with some particular value of E. We can label the solutions by an index n, so that each function n has a corresponding eigenvalue En. We must also be sure that our wavefunctions are normalizable; this means that n themselves must be normalizable. It is convenient to require that Z 1 2 j n(x)j dx = 1: (3.11) −∞ The solution to the Schr¨odingerequation can then be obtained by expanding the initial wavefunction in terms of the stationary states, X Ψ(x; 0) = cn n(x) (3.12) n for some coefficients cn. The full solution is then X −iEnt=¯h Ψ(x; t) = cn n(x)e (3.13) n An alternative to this procedure is to develop a numerical approach to solving the Schr¨odingerequation directly, without expanding in stationary states. It is to this task that we now turn. IV. SOLVING THE TIME-DEPENDENT SCHRODINGER¨ EQUATION In this section we will develop techniques for solving the full Schr¨odinger equation nu- merically. The first step is to introduce a grid of space points, separated by some distance ∆x, on which we will determine the wavefunction. We will also discretize in time, that is, evaluate the wavefunction only for a discrete set of times separated by some ∆t. Hence we n replace the function Ψ(x; t) with the discrete values Ψi , where i labels the space point and n labels the time value. Specifically, the grid points are located at xi = xmin + i∆x (4.1) 5 and the discrete times are given by (taking the initial time to be t = 0) tn = n∆t: (4.2) The next step is to approximate the derivatives in the Schr¨odingerequation as differences. These should approximate the corresponding derivatives well as long as ∆x and ∆t are \small enough." More precisely, the change in Ψ from one grid point to the next or one time step to the next should be small compared to Ψ itself. Now, an obvious approximation to the time derivative is @Ψ(x; t) Ψn+1 − Ψn ≈ i i (4.3) dt ∆t which becomes exact as ∆t ! 0. To develop a discrete approximation to the second deriva- tive with respect to x, imagine that we Taylor expand Ψ(x; t) about the point x (I will suppress the dependence on t here for clarity): @Ψ 1 @2Ψ Ψ(x + ∆x) = Ψ(x) + ∆x + ∆x2 + ··· (4.4) @x 2 @x2 where all derivatives are evaluated at the point x. Eq. (4.4) implies that @2Ψ Ψ(x + ∆x) + Ψ(x − ∆x) = 2Ψ(x) + ∆x2 + O(∆x4): (4.5) @x2 If we now drop the higher order terms as unimportant for sufficiently small ∆x, then this can be interpreted as an approximate formula for the second derivative: @2Ψ Ψ(x + ∆x) + Ψ(x − ∆x) − 2Ψ(x) ≈ : (4.6) @x2 ∆x2 This difference formula is analogous to, e.g., @Ψ Ψ(x + ∆x) − Ψ(x) ≈ (4.7) @x ∆x which gives a discrete approximation to the first derivative. Actually, it is more analogous to the \symmetric" first derivative: @Ψ Ψ(x + ∆x) − Ψ(x − ∆x) ≈ ; (4.8) @x 2∆x showing that there is actually quite a bit of freedom in how these approximations are con- structed. Both eqs. (4.7) and (4.8) approach the exact derivative as ∆x ! 0 and so either is 6 a perfectly valid approximation for finite ∆x.