A connection between quantum mechanical and classical adiabatic invariance is shown for the special case of a charged particle moving in a homogeneous, time-dependent magnetic field. The relationship consists of the fact that the transition probability between energy eigenstates corresponding to the magnnetic field at two different times is a function of a certain classical adiabatic invariant alone. From this relation follows the well-known result that during sufficiently slow changes of the magnetic field the particle will remain in one quantum state which passes continuously through the different instantaneous energies. The adiabatic invariance of this classical quantity is shown directly by solving the equations of niotion when the magnetic field changes linearly with time. (auth)
A connection between quantum mechanical and classical adiabatic invariance is shown for the special case of a charged particle moving in a homogeneous, time-dependent magnetic field. The relationship consists of the fact that the transition probability between energy eigenstates corresponding to the magnnetic field at two different times is a function of a certain classical adiabatic invariant alone. From this relation follows the well-known result that during sufficiently slow changes of the magnetic field the particle will remain in one quantum state which passes continuously through the different instantaneous energies. The adiabatic invariance of this classical quantity is shown directly by solving the equations of niotion when the magnetic field changes linearly with time. (auth)
We study the evolution of the energy and magnetic moment of a quantum charged particle placed in a homogeneous magnetic field, when this field changes its sign adiabatically. We show that after a single magnetic field passage through zero value, the famous adiabatic invariant ratio of energy to frequency is reestablished again, but with a proportionality coefficient higher than in the initial state. The concrete value of this proportionality coefficient depends on the power index of the frequency dependence on time near zero point. In particular, the adiabatic ratio of the initial ground state (with zero radial and angular quantum numbers) triplicates if the frequency tends to zero linearly as a function of time. If the Larmor frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of the time intervals between zero points, so that the mean energy behavior can be quasi-stochastic after many passages through zero value. The original Born-Fock adiabatic theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many instantaneous Fock states, whose weights do not depend on time in the new adiabatic regime.
Abstract The postulate of the existence of stationary states in multiply periodic dynamical systems requires that if the condition of such a system, when quantised, is changed in any way by the application of an external field or by the alteration of one of the internal constraints, the new state of the system must also be correctly quantised. It follows that the laws of classical mechanics cannot in general be true, even approximately, during the transition. There is one kind, of change, however, during which one may expect the classical laws to hold, namely, the so-called adiabatic change, which takes place infinitely slowly and regularly, so that the system practically remains multiply periodic all the time. In this case the quantum numbers cannot change, and it should be possible to deduce from the classical laws that the quantum integrals remain invariant. This was attempted by Burgers, who showed that they are invariant provided there are no linear relations of the type Σrmrωr = 0 (1) between the frequencies ωr, of the system, where the mr are integers. In general, however, the frequencies will alter during the adiabatic change, and in so doing will pass through an infinity of values for which relations such as (1) hold. A closer investigation is therefore necessary, as was pointed out by Burgers himself. In the following work, conditions which are rigorously sufficient to ensure the invariance of the quantum integrals, are obtained in such a form that it is possible for one to see whether they are satisfied or not without having to integrate the equations of adiabatic motion.
The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.
We outline a rather extraordinary series of similarities between classical and quantal behavior in the limit of adiabatic time changes. These include the power laws for the goodness of the respective invariants for isolated eigenstates and invariant tori for integrable systems, the nature of the breakdown of the invariances—level crossing in quantum systems and the role of ever present non-linear resonances is examined in the case of generically non-integrable classical dynamics—and the perhaps surprising relationship for fully chaotic systems where sufficiently slow switching in either classical or quantal systems precisely preserves the number of energy levels up to a given energy. For suitably small values of Planck's constant these similarities yield clear examples of the Bohr correspondence principle linking classical and quantum mechanics; for larger values the details in the classical picture are quenched in the quantum.
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.
We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential A=B(t)-y(1+α),x(1-α)/2, with two fixed values of the gauge parameter α: α=0 (the circular gauge) and α=1 (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for α=0 and straight lines for α=1. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function B(t). They are expressed in terms of solutions to the classical equation of motion ε¨+ωα2(t)ε=0, with ω1=2ω0. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions B(t), when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.