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Journal of Physics B Atomic and Molecular Physics

A unitary perturbation method for a paramagnetic atom in a magnetic field with constant and oscillatory components

F Ansbacher

Published Aug 1, 1973• 8 citations
Publisher

Abstract

The method is based on replacing the actual field B by an 'integrable' field, i.e. one for which the Schrodinger equation can be integrated. A simple criterion allows the best choice of field to be made for a given degree of approximation. Two examples are discussed: (1) the classic case of a small oscillating field transverse to a large steady field to an approximation beyond the rotating wave approximation, which reproduces, with certain modifications, the results of Bloch and Siegert valid for any j; (2) the field configuration of Pegg and Series (1970).

DOI
10.1088/0022-3700/6/8/039

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The method is based on replacing the actual field B by an 'integrable' field, i.e. one for which the Schrodinger equation can be integrated. A simple criterion allows the best choice of field to be made for a given degree of approximation. Two examples are discussed: (1) the classic case of a small oscillating field transverse to a large steady field to an approximation beyond the rotating wave approximation, which reproduces, with certain modifications, the results of Bloch and Siegert valid for any j; (2) the field configuration of Pegg and Series (1970).
A perturbation series is derived for the Schr\"odinger equation where the perturbation is permitted to go to \ensuremath{\infty}. Such an asymptotic series can be obtained in a representation where the time evolution of the states is due to the unperturbed part of the Hamiltonian of the problem and the time evolution of the observables is determined by the perturbation, contrary to the interaction picture. The equivalence between the series given by such a picture, and an asymptotic perturbation scheme is given. The method is applied to a spin-1/2 particle in a two-component magnetic field, one component of which, considered as a perturbation, can be time varying. Choosing a constant perturbation we show the equivalence between the exact solution and the approximate one when a component of the magnetic field is much larger than the other. In this case the limit t\ensuremath{\rightarrow}\ensuremath{\infty} cannot be taken, as mixed-secular terms appear in the asymptotic series. Taking a linear time-varying dependence for a component of the magnetic field, we get a nonanalytic asymptotic series.
Abstract The problem in question is to find the motion of an atom having arbitrary spin J in a strong oscillating magnetic field and a static field whose component perpendicular to the oscillating field is small (case B). It is shown how to relate this problem to that of ordinary magnetic resonance–an atom in a small oscillating field perpendicular to a strong static field (case A)–by applying to each case a suitable transformation. In the transformed frames the dominant terms in the equations of motion are equivalent and subsidiary terms are similar, so that results well-known in case A may be taken over to case B. The dominant terms in case B describe a set of resonances whose line-shapes are given explicitly for a particular example. Subsidiary terms lead to shifts of the resonances analogous to the Bloch-Siegert shift. The leading term s in these shifts are given. Terms of higher order may be calculated by employing a technique due to Shirley which is described.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
The time-dependent Schr\"odinger equation is solved formally for an atomic or molecular system which is subjected simultaneously to a rotating magnetic field of constant amplitude and angular velocity and to a constant magnetic field along the axis of rotation. The method yields the transition probabilities in terms of the solutions to an eigenvalue problem. This eigenvalue problem is solved both for (a) a normal Zeeman effect and for (b) the case where a transition from a given level is isolated in frequency from other transitions from the same level. Case (a) is exactly soluble and yields a solution which is shown to be the same as that of Bloch and Rabi, but is in a form which is more convenient for integration over the velocity distribution. Case (b) must be solved by an approximate method which results in a prediction of multiple quantum transitions as observed by Kusch.