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Physical Review

On Nonadiabatic Processes in Inhomogeneous Fields

Julian Schwinger

Published Apr 15, 1937• 119 citations
Publisher

Abstract

The problem of calculating nonadiabatic transition probabilities is considered. It is shown that the general G\"uttinger equations are incorrect and lead to erroneous results in any case other than that of the rotating magnetic field, which he considered. The corrected equations are applied in the calculation of the transition probabilities between the various magnetic states of a field precessing with constant angular velocity.

DOI
10.1103/physrev.51.648

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Progress of Theoretical Physics Supplement•Jan 1, 2002•0 citations
The problem of calculating nonadiabatic transition probabilities is considered. It is shown that the general G"uttinger equations are incorrect and lead to erroneous results in any case other than that of the rotating magnetic field, which he considered. The corrected equations are applied in the calculation of the transition probabilities between the various magnetic states of a field precessing with constant angular velocity.
Model calculations are presented for nonadiabatic passage of an oriented spin through a reversing solenoidal magnetic field; these are useful for comparison with experiments described elsewhere. The model corresponds closely to the well-known Landau-Zener-Stueckelberg model of a two-state curve crossing in molecular collision theory; various modifications appropriate to the Majorana problem are considered. The resulting transition amplitudes are discussed for the F=1F=1F=1 hyperfine component of the 22S122^{2}S_{\frac{1}{2}}22S21​​ metastable state of atomic hydrogen as an example.
The nonadiabatic transitions which a system with angular momentum JJJ makes in a magnetic field which is rotating about an axis inclined with respect to the field are calculated. It is shown that the effects depend on the sign of the magnetic moment of the system. We therefore have an absolute method for measuring the sign and magnitude of the moment of any system. Applications to the magnetic moment of the neutron, the rotational moment of molecules, and the nuclear moment of atoms with no extra-nuclear angular momentum are discussed.
The transitions between states of different space quantization which take place when an atom is passed through a weak inhomogeneous magnetic field are calculated by approximate methods taking into account the coupling between the nucleus and the orbital electrons. Two types of field are considered, namely, the field introduced by Majorana and the one introduced by G\"uttinger. For the Majorana field a discussion of the cases III arbitrary J=12J=\frac{1}{2}J=21​, JJJ arbitrary I=12I=\frac{1}{2}I=21​, and III and JJJ arbitrary with explicit application to the nitrogen atom is given. For the G\"uttinger field the general equations are set up for a field of any strength and are then applied to the case I=J=12I=J=\frac{1}{2}I=J=21​.
The probability amplitude for a spin-1/2 particle in a precessing magnetic field to be in an energy eigenstate is calculated exactly. The probability for the particle to be in a given energy eigenstate and the corresponding phase of the amplitude are obtained. In the adiabatic limit there are no transitions, and the phase is the sum of the dynamical phase and the Berry phase. When a unitary transformation is made, the Hamiltonian and the energy operator transform differently. The energy eigenvalues, Berry phases, and probability amplitudes calculated in a frame rotating with the magnetic field are the same as in the laboratory frame. If the energy operator is incorrectly taken to be the Hamiltonian in the rotating frame, incorrect energy eigenvalues, Berry phases, and probability amplitudes are obtained. Nevertheless, in the adiabatic limit the correct total phase results.
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.