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Transient phenomena in quantum mechanics

Salvador Godoy, Tobias Kramer, M. Moshińsky

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

Transient phenomena in quantum mechanics imply solving the time dependent Schrodinger equation with appropriate initial and boundary conditions. In this paper we consider the general formulation of the one dimensional problem and apply it to the particular example of transient phenomena for the bound state of a delta potential perturbed by the action of a boundary condition.

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Transient phenomena in quantum mechanics imply solving the time dependent Schrodinger equation with appropriate initial and boundary conditions. In this paper we consider the general formulation of the one dimensional problem and apply it to the particular example of transient phenomena for the bound state of a delta potential perturbed by the action of a boundary condition.
Transient phenomena in quantum mechanics have been of interest to one of the authors (MM) since long ago and, in this paper, we focus on the problem of a potential V - which for negative times gives rise to bound states and is suddenly changed at t = 0 to a potential V + which includes V -plus a perturbed term.An example will be the deuteron (where the proton and neutron are assumed to interact through an oscillator potential) submitted to a sudden electrostatic field.The analysis for t > 0 can be carried out with the help of appropriate Feynmann propagators and we arrive at the result that the separation between the nucleons has an amplitude that depends on the intensity of the electrostatic field, but its period continues to be related with the inverse of the frequency of the oscillator proposed for the interaction.A general approximate procedure for arbitrary problems of this type is also presented at the end.
The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as t−3t^{-3}t−3. The result is generalized for all square-integrable solutions to this problem.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.