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arXiv:2207.02198

Gauge- and coordinate-invariant equations for two-component systems

Ryan Requist

Published Jul 5, 2022• 0 citations
arXivHTMLPDF

Abstract

The Schrödinger-like equations for the marginal and conditional probability amplitudes resulting from the exact factorization of the wavefunction of a two-component system are derived in a form that is invariant to gauge and coordinate transformations. Coupled equations equivalent to the nonrelativistic Schrödinger equation of a molecule are derived as an example.

DOI
10.48550/arxiv.2207.02198

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The Schrödinger-like equations for the marginal and conditional probability amplitudes resulting from the exact factorization of the wavefunction of a two-component system are derived in a form that is invariant to gauge and coordinate transformations. Coupled equations equivalent to the nonrelativistic Schrödinger equation of a molecule are derived as an example.
Formulations of quantum mechanics incorporating the Weyl gauge transformations are studied in this article and developed further. In the process, impact of the method of observation on its outcome is interpreted in terms of the assigned gauges by incorporating properties of the corresponding experimental arrangement in defining them. Further, the assigned gauge is explicitly incorporated in the Feynman path integral formulation of quantum mechanics. The resulting wavefunction, which is not uniquely defined, represents a gauge equivalence class. The representative wavefunction is still obtained by the original path integral method. Methods to obtain the pertinent information about the assigned gauges supplementing the representative wavefunction are discussed. The probability density is shown to be a uniquely defined gauge invariant quantity but at the expense of some information describing the observable effects contained in gauge factors. In the standard quantum mechanics, a wavefunction is assumed to be defined within a phase factor while the probability density is phase-independent, paralleling these results. Also, the path integral method is used to deduce the Klein-Gordon equation for the representative wavefunction in the Riemannian spaces in a more streamlined manner than the previous derivations.
The wave function of quantum mechanics is not a boost invariant and gauge invariant quantity. Correspondingly, reference frame dependence and gauge dependence are inherited to most of the elements of the usual formulation of quantum mechanics (including operators, states and events). If a frame dependent and gauge dependent formalism is called, in short, a relative formalism, then the aim of the paper is to establish an absolute, i.e., frame and gauge free, reformulation of quantum mechanics. To fulfil this aim, we develop absolute quantities and the corresponding equations instead of the wave function and the Schrödinger equation. The absolute quantities have a more direct physical interpretation than the wave function has, and the corresponding equations express explicitly the independent physical aspects of the system which are contained in the Schrödinger equation in a mixed and more hidden form. Based on the absolute quantities and equations, events, states and physical quantities are introduced also in an absolute way. The formalism makes it possible to obtain some sharper versions of the uncertainty relation and to extend the validity of Ehrenfest's theorem. The absolute formulation allows wide extensions of quantum mechanics. To give examples, we discuss two known nonlinear extensions and, in close detail, a dissipative system. An argument is provided that the absolute formalism may lead to an explanation of the Aharonov-Bohm effect purely in terms of the electromagnetic field strength tensor. At last, on special relativistic and curved spacetimes, absolute quantities and equations instead of the Klein-Gordon wave function and equation are given, and their nonrelativistic limit is derived.
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.
, which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.
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.