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American Journal of Physics

Limits of the Adiabatic Approximation

Gianfranco Spavieri

Published Jun 1, 1971• 0 citations
Publisher

Abstract

The exact solution for the motion of a particle with charge e and mass m, moving in a spatially uniform magnetic field with the time variation B(t) = B0e−t/τ, is given in the nonrelativistic case. The trajectory of the particle has been plotted for various values of τ and limits are given for the validity of the “adiabatic approximation” and the “sudden approximation”.

DOI
10.1119/1.1986241

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The exact solution for the motion of a particle with charge e and mass m, moving in a spatially uniform magnetic field with the time variation B(t) = B0e−t/τ, is given in the nonrelativistic case. The trajectory of the particle has been plotted for various values of τ and limits are given for the validity of the “adiabatic approximation” and the “sudden approximation”.
The adiabatic motion of a charged, spinning, quantum particle in a two-dimensional (i.e. of constant direction) magnetic field is studied. A suitable set of operators generalizing the kinematical momenta and the guiding centre operators of a particle moving in a homogeneous magnetic field is constructed. This allows us to separate the two degrees of freedom of the system into a fast and a slow one which are, in the classical limit, the rapid rotation of the particle around the guiding centre and the slow guiding centre drift. In terms of these operators the Hamiltonian of the system can be rewritten as a power series in the magnetic length , and the fast and slow dynamics separates. The effective guiding centre Hamiltonian is obtained to second order in the adiabatic parameter and reproduces correctly the classical limit.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
Expansions of a plane wave in Landau states are derived corresponding to an adiabatic and to a sudden switching on of the magnetic field. Two earlier solutions given by Faisal (1982) and by Ohsaki (1983) are found to be valid alternative expansions in the adiabatic case, which differ only in the choice of normalisations of the basis states used. A result identical to Ohsaki's is reobtained, since the Landau states are confined by conservation of energy to a large but finite area L x L y approximately 1/ gamma , perpendicular to the field. This energy conservation requires the corresponding sum over the degenerate Landau states to be restricted within a finite range. The upper and lower limits of this range are derived. The completeness relation of Faisal's expansion shows that it is also effectively and automatically confined to the same range of states. In the limit gamma to 0 both sets of expansion coefficients become identical and the results go over to the desired plane-wave state. In the sudden case, when the field switches instantaneously from zero to a finite value, the expansion no longer consists of Landau states of a fixed energy, and the result is new. The two results correspond to different limiting cases of any experimental situation, and the sudden expansion is shown to be invalid at the highest laboratory field ( approximately 20 kG) in current use. The experimental situations corresponding to the two different expansions are discussed.
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.