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European Journal of Physics

Conventional and gauge-invariant probability amplitudes when electromagnetic potentials are turned on and off

Donald H. Kobe

Published Jul 1, 1984• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/5/3/009

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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.
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 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 wavefunction of a quantum mechanical system can be evaluated in any gauge and can be expressed in terms of a complete set of eigenstates of a physical observable or of a mathematical operator. If the physical observable is expressed in the same gauge as the Hamiltonian from which the wavefunction is obtained, the expansion coefficients are gauge invariant and have definite experimental implications. This physical observable need not be the energy. If the basis set of vectors are eigenstates of a mathematical operator, the expansion coefficients are gauge dependent. This mathematical operator can be translated to a physical observable once the gauge under which the wavefunction is calculated is specified. The expansion coefficients can then be interpreted as probability amplitudes.
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.