Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Questions of gauge and basis sets for the representation of electromagnetic interactions in quantum mechanics

Silviu Olariu, Iovitzu Popescu, C. B. Collins

Published Dec 15, 1979• 14 citations
Publisher

Abstract

Recent concerns about the gauge dependence of transition amplitudes describing the changes of state of quantized systems interacting with electromagnetic radiations have led to the development of invariant schemes for the expression of the time-dependent quantities involved. It is shown in this work that such schemes are not uniquely specified by the requirement for gauge invariance. Moreover, they simply exchange a dependence upon the gauge of the electromagnetic potentials for a previously unrecognized dependence upon the representation of basis states describing initial and final states of the process that is equivalent to a dependence upon gauge. However, general expressions are developed in this work which show that resonant transitions which correspond to physically observable processes involving the pure or mixed absorption and emission of $N$ photons are gauge invariant regardless of the multipolarity of the transition moments and independent of the representation of the basis states of the system. A preferred representation is introduced which is shown to accommodate the physical effects generally encountered.

DOI
10.1103/physrevd.20.3095

Related papers

On gauge invariance in quantum transition amplitudes

C. K. Au

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.

Journal of Physics B Atomic and Molecular Physics•Feb 14, 1984•8 citations

Gauge invariant formulation of the interaction of electromagnetic radiation and matter

Donald H. Kobe, Arthur L. Smirl

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.

American Journal of Physics•Jun 1, 1978•118 citations

The physical significance of gauge independence and gauge covariance in quantum mechanics

T. E. Feuchtwang, Emil Kazes, Paul H. Cutler +1 more

The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.

Journal of Physics A Mathematical and General•Jan 11, 1984•15 citations

Operator Gauge Transformations in Nonrelativistic Quantum Electrodynamics

Raymond Dale Gray

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.

•Dec 1, 1982•0 citations

Gauge invariance in non–relativistic electrodynamics

R. Guy Woolley

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.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 8, 2000•47 citations

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

Kuo‐Ho Yang

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.

Journal of Physics A Mathematical and General•Apr 1, 1983•11 citations
Recent concerns about the gauge dependence of transition amplitudes describing the changes of state of quantized systems interacting with electromagnetic radiations have led to the development of invariant schemes for the expression of the time-dependent quantities involved. It is shown in this work that such schemes are not uniquely specified by the requirement for gauge invariance. Moreover, they simply exchange a dependence upon the gauge of the electromagnetic potentials for a previously unrecognized dependence upon the representation of basis states describing initial and final states of the process that is equivalent to a dependence upon gauge. However, general expressions are developed in this work which show that resonant transitions which correspond to physically observable processes involving the pure or mixed absorption and emission of NNN photons are gauge invariant regardless of the multipolarity of the transition moments and independent of the representation of the basis states of the system. A preferred representation is introduced which is shown to accommodate the physical effects generally encountered.
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 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 distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.
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.
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.
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.