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Journal of Physics A Mathematical and General

Gauge invariance and the space-translation method

Donald H. Kobe

Published Jan 11, 1984• 7 citations
Publisher

Abstract

The space-translation method for treating the interaction of electromagnetic radiation and matter in the electric dipole approximation is investigated from the standpoint of gauge invariance. Unless the wavefunction of the space-translation method is properly transformed, the probability of finding the system in an energy eigenstate is not in general correct. In the electric dipole approximation the probability of finding the system in an energy eigenstate is most conveniently calculated in the electric field gauge where the interaction is of the length form. The length form of the interaction is not in general equal to the velocity form or the acceleration form which comes from the space-translation method. The charged harmonic oscillator is used as an example.

DOI
10.1088/0305-4470/17/1/016

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The space-translation method for treating the interaction of electromagnetic radiation and matter in the electric dipole approximation is investigated from the standpoint of gauge invariance. Unless the wavefunction of the space-translation method is properly transformed, the probability of finding the system in an energy eigenstate is not in general correct. In the electric dipole approximation the probability of finding the system in an energy eigenstate is most conveniently calculated in the electric field gauge where the interaction is of the length form. The length form of the interaction is not in general equal to the velocity form or the acceleration form which comes from the space-translation method. The charged harmonic oscillator is used as an example.
The gauge-invariant formulation of quantum mechanics is applied to the charged harmonic oscillator in an electromagnetic field in the electric dipole approximation to obtain the probability that the oscillator is in a particular state at time t. This probability is compared with the corresponding 'probability' calculated from the conventional approach to the interaction of radiation with matter using the interaction A.p. The probabilities do not agree with each other. From the principle of gauge invariance it is concluded that the probability calculated from the gauge-invariant formulation is correct in this case, and that the conventional approach in general is incorrect.
A manifestly gauge-invariant formulation of quantum mechanics is applied to a charged isotropic harmonic oscillator in a time-varying magnetic field in the magnetic dipole approximation. The energy operator for the problem is the sum of the kinetic and potential energies. The kinetic energy operator is the square of the gauge-invariant kinetic momentum operator divided by twice the mass. The energy eigenvalues and state probabilities are calculated and are shown to be the same in all gauges. In this problem there is no gauge in which the energy operator reduces to the unperturbed Hamiltonian, as there is in the electric dipole approximation. Consequently, eigenvalues of the unperturbed Hamiltonian and corresponding (gauge-dependent) state 'probabilities' are different from the gauge-invariant quantities in all gauges.
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 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.