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American Journal of Physics

Spreading of wave packets in the coordinate representation

Chai Shing Lai

Published Sep 1, 1979• 4 citations
Publisher

Abstract

By dealing with the Schrödinger equation for a free particle in the coordinate representation alone, the expression for ψ (r,t) at time t is derived in terms of ψ (r,t0) at an initial instant t0 with the use of the ’’method of self-similar solution of partial differential equations.’’ The use of Green’s function is necessary due to the complex nature of the Schrödinger equation. The same technique can also be applied to the case of spreading of wave packets in a dissipative medium.

DOI
10.1119/1.11924

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By dealing with the Schrödinger equation for a free particle in the coordinate representation alone, the expression for ψ (r,t) at time t is derived in terms of ψ (r,t0) at an initial instant t0 with the use of the ’’method of self-similar solution of partial differential equations.’’ The use of Green’s function is necessary due to the complex nature of the Schrödinger equation. The same technique can also be applied to the case of spreading of wave packets in a dissipative medium.
The solution is essentially a complex function which accounts for the matter wave spreading illustrated in Figure 28.3. As the wave function itself cannot be identified with a physical property of the free particle, an indirect interpretation was suggested by Born. He pointed out that its modulus ( tx,Ψ ) ( ),x tΨ is large where the particle becomes localized and small elsewhere, and hence ( ) ( ) 2, ,P x t dx x t dx= Ψ (29.1) is the probability of finding the particle described by ( )tx,Ψ at a point between x and at a time t. Therefore the matter wave spreading is associated with a decrease in the probability density . As the total probability of finding the particle somewhere on the x-axis must be unity, the wave function must be normalized to unity, that is dxx + ( txP , ) ( ) ( ) ( ), , ,P x t dx x t x t dx∞ ∞ ∗ −∞ −∞ = Ψ Ψ∫ ∫ = (29.2) The behaviour of the wave function ( )tx,Ψ is determined by Eq.(28.38), which may be rewritten as ( ) ( ) ( ) t x tx m itxttx ∆∂ Ψ∂+Ψ=∆+Ψ 2 2 , 2 ,, = (29.3) where the expression ( ) ( )/ , ,t x t t x t⎡∂Ψ ∂ ≅ Ψ + ∆ −Ψ ∆⎣ / t⎤⎦ ) , which holds for a small , has been used. In view of Eq.(29.3) the values of the wave function at any time can be derived from the initial value that describes a reference state of the particle.
Standard eikonal methods, as asymptotic perturbation expansions in 1/\ensuremath{\Elzxh}, are incapable of describing fully the dispersive spreading of a wave packet. By simple models, we distinguish the two types of dispersion associated with a packet having initially uniform and nonuniform distributions of wave number. A nonperturbative extension of the eikonal approximation is applied to two initial-value problems for the free-particle Schr\"odinger equation and compared with exact solutions. The method maintains the ray picture fully. Applications of the method to more general classes of problems are discussed.
We have seen that to determine the possible energies En (energy eigenvalues) which a physical system can have and the corresponding eigenfunctions (wave functions) ψn, we have to set up and solve the time-independent Schro¨dinger equation Hψn = Enψn . (8.1.1) In this equation, the wave function ψn, representing the n-th eigenstate of the system in the coordinate representation, is a function of the continuously varying eigenvalues of the position observables of the system, the number of position observables being equal to number of degrees of freedom of the system. The Hamiltonian operator H, in the coordinate representation, is a differential operator involving the continuously varying eigenvalues of the position observables and the derivatives with respect to them. Since we will be dealing with the operators mostly in the coordinate representation, we shall denote them without the caret.
The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.