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Journal of Physics A Mathematical and General

Gauge-invariant time-dependent perturbation theory. II. degenerate case

Kuo‐Ho Yang

Published Apr 1, 1983• 11 citations
Publisher

Abstract

For pt.I see ibid., vol.15, p.1201 (1982). The author formulates a manifestly gauge-invariant time-dependent perturbation theory for a nonrelativistic spinless charged particle interacting with a classical external electromagnetic radiation field. The treatment of the degenerate bound states and the states in the continuum is considered. It is found that the gauge-invariant and the conventional perturbative solutions agree on the net transition rates but disagree on the transition amplitudes and transition probabilities. The treatment of a realistic physical system can be obtained by combining the treatment here with that for a nondegenerate system.

DOI
10.1088/0305-4470/16/5/012

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For pt.I see ibid., vol.15, p.1201 (1982). The author formulates a manifestly gauge-invariant time-dependent perturbation theory for a nonrelativistic spinless charged particle interacting with a classical external electromagnetic radiation field. The treatment of the degenerate bound states and the states in the continuum is considered. It is found that the gauge-invariant and the conventional perturbative solutions agree on the net transition rates but disagree on the transition amplitudes and transition probabilities. The treatment of a realistic physical system can be obtained by combining the treatment here with that for a nondegenerate system.
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 Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.
A novel Hamiltonian scheme for non–relativistic quantum electrodynamics in which the gauge arbitrariness of the field potential is kept explicit is used to study the gauge–dependence properties of various versions of perturbation theory when resonance and line–broadening effects are admitted. Time–dependent perturbation theory is shown to have severe gauge–dependence problems unless ad hoc modifications are made. A time–independent formulation of S–matrix theory is then studied. Far from resonance, the S–matrix is gauge invariant in all orders of perturbation theory due to a very precise cancellation of gauge–dependent terms which requires, among other things, complete sets of intermediate states; energy conservation also has a crucial role. However, an obvious separation of the S–matrix into a resonant (pole) and non–resonant background leads to incomplete cancellation of the gauge–dependent terms. The introduction of the Heitler damping matrix into an integral equation for the T–matrix leads to a gauge–invariant result. This provides the basis for a gauge invariant S–matrix theory of atoms and molecules interacting with electromagnetic radiation that encompasses resonance and damping effects.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
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.