Manifest Gauge Invariance in Nonrelativistic Quantum Mechanics with Classical Electromagnetic Fields

Abstract

Since gauge invariance is such a fundamental symmetry principle, it is surprising that a manifestly gauge-invariant formulation of nonrelativistic quantum mechanics did not emerge until a few years ago. In 1976 Yang1 gave such a formulation in which he took as physical observables only Hermitian, gauge-invariant operators.2,3 For a charged particle in an external classical electromagnetic field, the Hamiltonian is not in general the energy operator. An energy operator is defined as the sum of the kinetic and potential energies, which is the Hamiltonian minus the scalar potential of the external time-dependent electromagnetic field.4–6 The eigenvalue equation for the energy operator gives the energy eigenvalues and energy eigenstates at any time. The probability amplitude for finding the particle in an energy eigenstate is the inner product of the eigenstate of the energy operator and the wave function. This amplitude is gauge invariant, and therefore can be used to obtain a valid probability. The energy eigenstates are coupled by the matrix elements of the quantum-mechanical power operator.7