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European Journal of Physics

On the link between Schrodinger and Heisenberg pictures of quantum mechanics

Peter Prešnajder, V. Pažma

Published Jan 1, 1989• 0 citations
Publisher

Abstract

The authors take into account boundary conditions in solutions of the Heisenberg equations of motion. They express the kernel of the evolution operator in the x-representation through the retarded propagator which solves the Schrodinger equation and satisfies the boundary conditions in question. As an illustration, they solve the Heisenberg equation for a free particle bound to the half-line (0, infinity ) with various boundary conditions at the point x=0.

DOI
10.1088/0143-0807/10/1/008

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The authors take into account boundary conditions in solutions of the Heisenberg equations of motion. They express the kernel of the evolution operator in the x-representation through the retarded propagator which solves the Schrodinger equation and satisfies the boundary conditions in question. As an illustration, they solve the Heisenberg equation for a free particle bound to the half-line (0, infinity ) with various boundary conditions at the point x=0.
The origin and the physical interpretation of boundary conditions for a free particle confined to a half-line are analysed. The authors consider potentials V(x,a, . . .) (depending upon parameters a, . . . and x in (- infinity , + infinity )) for which solutions psi (x,a, . . ., t) of the corresponding Schrodinger equation reduce, in a suitable limit a from a 0 , . . ., to solutions psi (x,t) of the Schrodinger equation for a free particle confined to a half-line satisfying the boundary condition ( delta psi / delta x)(x=0, t)=a psi (x=0, t) with real alpha = alpha (a 0 , . . .).
It is well known that the motion of a dynamical system can be pictured in two distinct ways, which Dirac names the Heisenberg picture and the Schrödinger picture. The equations of motion take different forms in the two pictures, but of course have identical physical consequences, since the motion of the system does not depend in any way on which picture we choose. It should therefore be possible to express the equations in an invariant form (independent, that is, of the picture used). It will be shown in this note that not only can this be done (equation (3)), but it can be done in such a way that it is not even necessary to introduce a picture at all (equation (3′)).
Some years have passed since the appearance of Dirac's striking interpretation of the Heisenberg-Schroedinger parallelism in the equations of motion, and yet it appears that there still remain obscure points which have led to misunderstanding. Although the Schroedinger method suffices to deal with many practical problems, the flexibility of the Dirac-Jordan representation theory, aside from its intrinsic unity and beauty, recommends it to general theoretical investigation. The present paper considers the invariant form of the equations of motion in relation to the Schroedinger-Heisenberg pictures of quantum mechanics insofar as the latter refer to the "pictured" intrinsic motion of states (vectors) and observables (Hermitian operators). The discussion is divided roughly into 5-sections: Operator and Vector Transforms, Contact Transformations, Heisenberg-Schroedinger Operators, Equations of Motion and Invariant Properties, and the Density Operator and Operator Spur. Where possible, quasi-geometrical diagrams are given to illustrate the relation and transition between the several "pictures of the motion."
For many problems a solution in the Heisenberg picture is straightforward, but the Schrodinger picture solution is difficult, although presumably it is possible. A good example is the driven harmonic oscillator. The equations of motion are easy to solve when the driving force is periodic. The Hamiltonian can be reduced to time-independent form and the evolution operator is easy to write down as a single exponential. To work out the transition probability from one number state to another, it is necessary to decompose the exponential. As well as the Baker-Hausdorff formula, other operator techniques are used together with the coherent-state projection operator, to solve the problem.
The Schrödinger equation for the scattered particle is converted into an integral equation, in which the kernel is obtained by the use of the condition that the events described by the solutions of the equation must be causally related, i.e., that causes must always precede the effects which they produce. The integral equation is iterated once, and it is found that its solution corresponds to the usual description of a scattering process in terms of an undeflected wave traversing the system plus a radially scattered wave moving outwards. In the second part, it is shown that a simple relation exists between solutions of Schrödinger's equation of the sort just described and the solution of the initial-value problem associated with the equation of motion for a free particle. By this means a solution of the scattering problem is set up which describes how the wave function develops as the wave which it describes propagates through the system. The series which results was first given by Feynman. As an example, the first Born approximation for the scattering problem is developed by each approach.