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Journal of Physics B Atomic and Molecular Physics

An improved coordinate transformation for the treatment of a paramagnetic atom in a linear oscillatory and a constant transverse magnetic field

F Ansbacher

Published Aug 1, 1973• 3 citations
Publisher

Abstract

The introduction of a suitably moving coordinate system results in approximate equations of motion which lead in a direct manner to the Block-Siegert resonance shift, to 2nd or higher order, and to a simple visualization of multiple quantum transitions, including their correct quantitative treatment. The relation between the present method and that due to Shirley (1965), whose results are confirmed, is pointed out.

DOI
10.1088/0022-3700/6/8/040

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Physics and Time-Space

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In this paper we propose an idea to put the wave functions describing the move states of particles into a coordinate transformation and than point out it may be good for treatment of some quantum problems.

Journal of Liaoning University•Jan 1, 2011•0 citations
The introduction of a suitably moving coordinate system results in approximate equations of motion which lead in a direct manner to the Block-Siegert resonance shift, to 2nd or higher order, and to a simple visualization of multiple quantum transitions, including their correct quantitative treatment. The relation between the present method and that due to Shirley (1965), whose results are confirmed, is pointed out.
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 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 semi-classical approach developed by Pegg and Series (1970, 1973) is used to study higher order effects in magnetic resonance produced by an oscillating magnetic field perpendicular to a static field as the oscillating field strength is increased. It is shown that the well-known 'multiple quantum' transitions arise from a semi-classical theory, being transformed into 'single quantum' transitions induced by field harmonics in the most appropriate reference frame for treating the problem. Higher order terms in the Bloch-Siegert shift support those derived by Shirley (1965) by means of Floquet theory, and are in gross disagreement with those derived by Chang and Stehle (1971) from quantum electrodynamics, particularly at very high field intensities where the results are not even qualitatively similar. It should be possible to decide easily between the two approaches by experiment.