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Journal of Mathematical Physics

A time-dependent Schrödinger equation

H. U. Kranold

Published Sep 1, 1983• 2 citations
Publisher

Abstract

We reduce the solution of the Schrödinger equation with the potential U(r, t)=α1(t) r2+α2(t) x+α3(t) y +α4(t) z+α5(t) to the solution of the Schrödinger equation for a free particle. The αi(t) are arbitrary functions of time. A generalization of this is also considered.
DOI
10.1063/1.525983

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We solve the Schrodinger equation analytically for systems whose potentials have a certain time-dependence (which is monotonously decreasing) and general coordinate-dependences.Only a few time-dependent systems have been reported to be analytically solved whose potentials are constant, linear, and quadratic functions of coordinate with arbitrary time-dependences.From a different perspective, we focus on the time-dependent systems whose potentials are monotonously decreasing functions of time with arbitrary coordinate dependences.Time-dependent potential of any coordinate-dependence can be handled analytically by transforming it to a time-independent potential of known solutions if its time-dependence is monotonously decreasing.We do this by a unitary transformation of the wavefunction and variable transformations to change the Schrodinger equation to be time-independent in new variables.These variables are then determined by solving a set of simple differential equations.This way we are able to find and to obtain analytical solutions for time-dependent potentials which we mention above.
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).
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 , . . .).