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Acta Scicentiarum Naturalum Universitis Pekinesis

Adiabatic Switching of Electromagnetic Fields and the Energy Operator

Qian Shang-Wu

Published Jan 1, 1990• 0 citations
OpenAlex

Abstract

When the electromagnetic field is adiabatically switched on and subsequently switched off, the vector and scalar potentials can still be nonzero when the electromagnetic fiele is zero. In this case it is necessary to use the energy operator, not the Hamiltonian, to calculate the probability that the system is in an energy eigenstate. In the gauge in which the potentials are zero when the field is zero, the energy operator reduces to the unperturbed Hamiltonian and the usual procedure for calculating probabilities can be used.

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When the electromagnetic field is adiabatically switched on and subsequently switched off, the vector and scalar potentials can still be nonzero when the electromagnetic fiele is zero. In this case it is necessary to use the energy operator, not the Hamiltonian, to calculate the probability that the system is in an energy eigenstate. In the gauge in which the potentials are zero when the field is zero, the energy operator reduces to the unperturbed Hamiltonian and the usual procedure for calculating probabilities can be used.
The amplitude for finding a particle in an unperturbed state is equal to the gauge-invariant probability amplitude for finding the particle in an energy eigenstate when the vector potential is turned off. The artificial practice in quantum mechanics of turning the vector potential on and off leads under some circumstances to unrealistic electromagnetic fields. For sudden switching of the potentials, a realistic electromagnetic field is obtained only when the vector potential and electromagnetic field are zero at the time of switching on and off. For the adiabatic case, a realistic electromagnetic field is obtained only in the limit when the potentials are turned on in the infinite past and turned off in the infinite future.
The case of a single charged particle with potential energy is considered when the classical electromagnetic field is zero. The vector and scalar potentials describing this situation are not necessarily zero. When the conventional approach to radiation problems using the interaction A⋅p plus p⋅A plus A2 plus A0 (constants omitted) is used, time-dependent probabilities are predicted if the vector potential A=−∇Λ≠0 even though the electromagnetic field is zero. The reason for this unphysical behavior is that eigenfunctions of the unperturbed Hamiltonian, which is not gauge invariant, are used to calculate the probability amplitudes. When a manifestly gauge-invariant procedure is applied to this case the probabilities are constant in all gauges. The reason that physical results are obtained is that eigenstates of the energy operator are used to calculate the probability amplitudes.
The conventional perturbation theory in quantum mechanics for the interaction of electromagnetic radiation with matter, which is based on the time-dependent vector and scalar potentials, is shown to be dependent on the gauge. The problem is reformulated, using the approach of Yang, in a gauge-invariant way, in which the transitions between states are induced by the quantum-mechanical power operator. In order to use an unperturbed Hamiltonian which does not involve the electromagnetic field, the gauge is chosen to give the electric dipole interaction. This choice can be made if the magnetic field is negligible and the electric field is slowly varying over atomic dimensions. For a comparison of the gauge-invariant and conventional formulation, a two-level atom in a radiation field is considered. For the probability that an atom originally in the lower state will make a transition to the upper state when the field is not in resonance, the gauge-invariant formulation gives the correct result, whereas the gauge-dependent formulation gives an incorrect result.
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.