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arXiv:quant-ph/0511212 · Symmetry Integrability and Geometry Methods and Applications

Transient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation

M. Moshińsky

Published Aug 18, 2005• 1 citation
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Abstract

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.

DOI
10.3842/sigma.2005.003

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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.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
The subject of transient effects in quantum mechanics has been of interest to one of the authors (MM) since long before it became possible to study them experimentally. In particular, the problem of opening a shutter instantaneously led him to the concept of diffraction in time [1]. Physically, it is only possible to open a shutter as a function of time and this complicates the problem greatly, as it is then not invariant under time translations and thus, among other difficulties, the energy of the system is not a constant of the motion. Kleber and Scheitler [2] analyzed the problem describing the shutter as a ±¡function potential at the origin x = 0, but whose strength was an inverse function of time. In this paper, we follow another procedure by adding to our initial two particle channel another one, and make them interact through appropriate time independent boundary conditions at the point of coincidence. The full problem conserves the total energy, but this does not happen if we restrict ourselves to a description in the first channel only. Thus, we have a rough but analytically solvable problem using a Laplace transform, which mimics some aspects of a time-dependent shutter, and compare our results with some of those derived by the different procedure of reference [2].
We study the behavior of a quantum particle trapped in a confining potential in one dimension under multiple sudden changes of velocity and/or acceleration. We develop the appropriate formalism to deal with such situation and we use it to calculate the probability of transition for simple problems such as the particle in an infinite box and the simple harmonic oscillator. For the infinite box of length L under two and three sudden changes of velocity, where the initial and final velocity vanish, we find that the system undergoes quantum revivals for Δt=τ0≡4mL2πℏ, regardless of other parameters (Δt is the time elapsed between the first and last change of velocity). For the simple harmonic oscillator we find that the states obtained by suddenly changing (one change) the velocity and/or the acceleration of the potential, for a particle initially in an eigenstate of the static potential, are coherent states. For multiple changes of acceleration or velocity we find that the quantum expectation value of the Hamiltonian is remarkably close (possibly identical) to the corresponding classical expectation values. Finally, the probability of transition for a particle in an accelerating harmonic oscillator (no sudden changes) calculated with our formalism agrees with the formula derived long time ago by Ludwig [Z. Phys. 130(4), 468–475 (1951)], and recently modified by Dodonov [J. Russ. Laser Res. 42(3), 243–249 (2021)], but with a different expression for the dimensionless parameter γ. Our probability agrees with the one of Dodonov for γ ≪ 1 but is not periodic in time (it decays monotonously), contrary to the result derived by Dodonov.
This chapter discusses the time-dependent perturbation theory applicable for small time-dependent perturbation. The incident electromagnetic wave makes the electric and magnetic potentials to oscillate and the perturbation is usually very small compared with the static Hamiltonian. The theory of time-dependent perturbation was developed by Dirac in 1927. Approximation methods are developed for systems where the time dependence of the Hamiltonian is small, slowly varying or rapidly varying. These correspond to perturbation, adiabatic and sudden approximations. An interesting consequence of sudden approximation is that the system can undergo an adiabatic change due to the sudden approximation. The rate of spontaneous emission is proportional to the number of atoms in the excited state. In the stimulated emission, an incident radiation of appropriate frequency induces an atom in an excited state to emit radiation. In the case of time-dependent perturbations, the system undergoes transition from one state to another during the action of the perturbation.