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Acta Scicentiarum Naturalum Universitis Pekinesis

The Unified Form of Constructing Gauge Independent Transition Probabilities for Interaction of Radiation with Matter

Qian Shang-Wu

Published Jan 1, 1990• 0 citations
OpenAlex

Abstract

It is pointed ont that there are two kinds of methods, to construct gauge inde-pendent transition probabilities (GITP), one kind corresponds to active view of gauge invariance in quantum mechanics, another kind corresponds to the passive view. Two kinds of view are interchangeable by means of unitary transformation. From the active view, all known methods of constructing GITP consist in choosing a gauge dependent unitary operator which transforms the non-physical quantity H0 into a true physical quantity to be measured, where H0 is the unperturbed Hamiltonian in the conventional formulation for interaction of radiation with matter.

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It is pointed ont that there are two kinds of methods, to construct gauge inde-pendent transition probabilities (GITP), one kind corresponds to active view of gauge invariance in quantum mechanics, another kind corresponds to the passive view. Two kinds of view are interchangeable by means of unitary transformation. From the active view, all known methods of constructing GITP consist in choosing a gauge dependent unitary operator which transforms the non-physical quantity H0 into a true physical quantity to be measured, where H0 is the unperturbed Hamiltonian in the conventional formulation for interaction of radiation with matter.
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 conventional formulation for interaction of radiation with matter uses a zero potential Hamiltonian H 0 as an unperturbed Hamiltonian and this procedure leads to a conflict between the gauge invariance of probability amplitudes and the Born rule. It is pointed out that if one uses a dark Hamiltonian H d as the unperturbed Hamiltonian and considers its gauge dependence, then this conflict no longer exists. From this point of view the author proves that the Lamb gauge and the Coulomb gauge in the electric dipole approximation give the same gauge-independent probability amplitudes, and thus the A.p against r.E controversy becomes unimportant.
For a quantum mechanical particle in a classical electromagnetic field, the active and passive views of gauge invariance are compared. In the active point of view, both the wavefunction and the potentials are transformed so that the Schrödinger equation is form invariant. In the passive point of view, a change in the representation of the canonical momentum is made which just cancels the change in the gauge of the potentials. The condition that a gauge-invariant operator has the same expectation value in all gauges gives a functional equation, the solution of which shows that gauge-invariant operators depend only on the kinetic momentum or coordinates.