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Physics Letters A

Gauge invariance and the virial theorem for the quantized Schrödinger field

Ken-ichi Takiuchi, Tadashi Toyoda

Published Oct 1, 1999• 3 citations
Publisher
DOI
10.1016/s0375-9601(99)00674-x

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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 gauge-invariant periodic quantization (GIPQ) method is applied within the complete Hilbert space of solutions of the exact time-dependent Schr\"odinger equation. The multiplicity of such GIPQ solutions is shown to exceed by far the number of eigenfunctions they are intended to analogize. However, time averaging (or the equivalent averaging over all values of the progressive phase) reduces in this space to a projection upon the exact eigenfunctions under which every one of these GIPQ solutions either vanishes identically or reduces precisely to one of the stationary (but unnormalized) eigenfunctions of the exact Hamiltonian. One concludes that the correspondence between the GIPQ solutions and exact eigenfunctions is well manageable in the complete Hilbert space, a result which encourages the further study of time-averaged GIPQ solutions in parametric subspaces, where time averaging projects not upon an eigenstate, but upon the constant component of the GIPQ periodic solution.
The local phase-invariance of the momentum-space Schrödinger equation for free-particle has been used to construct quantum kinematics that describes a motion of the particle in external U(1) background gauge field. The gauge structure over the momentum space of the particle is interpreted in terms of helicity and spin carried by the particle. As a by-product an effective one-particle Schrödinger equation of motion for the helicity-carrying particle in external potential field is derived. An effect of screening of the external potential is predicted, that can affect quantization and splittings of energy levels of the particle.