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Physical Review A

Alternative effective Hamiltonians in nonrelativistic quantum electrodynamics

G. Compagno, Edwin Albert Power

Published Oct 1, 1988• 5 citations
Publisher

Abstract

We consider, in the context of nonrelativistic quantum electrodynamics, the transformation from the minimal-coupling Hamiltonian to one where the interaction is quadratic in the electromagnetic vector potential. The new effective Hamiltonian is used to obtain the energy shifts in an atom caused by its interaction with the electromagnetic field. This demonstration is very compact since all the terms contributing to the shift arise from the single quadratic-coupling Hamiltonian. A brief comparison with other Hamiltonians having alternative couplings is made.

DOI
10.1103/physreva.38.4340

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Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Jun 4, 1974•93 citations
We consider, in the context of nonrelativistic quantum electrodynamics, the transformation from the minimal-coupling Hamiltonian to one where the interaction is quadratic in the electromagnetic vector potential. The new effective Hamiltonian is used to obtain the energy shifts in an atom caused by its interaction with the electromagnetic field. This demonstration is very compact since all the terms contributing to the shift arise from the single quadratic-coupling Hamiltonian. A brief comparison with other Hamiltonians having alternative couplings is made.
The electromagnetic field operators and the electron field operators for the coupled system governed by the multipolar Hamiltonian are obtained within the Heisenberg picture. Their causal behavior and their relationship to the minimal-coupling forms are discussed. The basic fields of the multipolar theory, namely, the displacement vector and the magnetic field, are calculated up to terms quadratic in the multipole moment sources. The terms linear in the transition moments are the quantum counterparts of the classical fields and do not change the photon occupation number. The quadratic terms have no classical analogs: they act in both the photon and electron occupation-number spaces. It is shown that it is necessary to include these second-order terms in the calculation of the Poynting vector for an emitting dipole, thus demonstrating their role in the transport of radiative energy.
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.
Introduction . In the previous chapter we have obtained the form of the coupling between matter and the electromagnetic field. We are thus in a position to treat electrodynamics in the presence of charges and currents. We start by obtaining the Euler-Lagrange equations of the matter-electromagnetic field system. These turn out to be the Maxwell-Lorentz equations, which is a set of coupled equations of motion in which the matter field acts as a source for the electromagnetic field and vice versa. In Section 4.2 we derive the Hamiltonian of the complete system in the Coulomb gauge, and this leads to the so-called minimal coupling Hamiltonian, containing both the electromagnetic potential and the matter-field amplitude. Specializing the nonrelativistic matter field to the case of a neutral atom, consisting of the field of electrons in a static nuclear potential, in Section 4.3 we obtain the atom-photon Hamiltonian in the minimal coupling scheme. Some of the basic processes induced by the atom-photon interaction part of the Hamiltonian are also discussed in this section. This gives us the possibility of introducing at this stage a fundamental simplification of the interaction Hamiltonian, namely the electric dipole approximation. The minimal coupling scheme, however, is not the only possible atom-field coupling.
The quantized electromagnetic field in interaction with nonrelativistic sources, i.e., atoms and molcules, has been the subject of study in the previous papers in this series. The Heisenberg formalism is used to calculate the electric- and magnetic-field operators arising from charges and currents expressed in multipolar form. In this paper, quadratic operators in these fields such as the Thompson energy density and the Poynting vector and their expectation values for specified states are found. The expectation values are, in general, time dependent. For initial conditions given at t=0 it is shown that the fields are causal, i.e., for tr/c the source-dependent fields are zero and the quadratic operators have only their zero-point contributions. For t>r/c they have both time-independent and time-dependent terms. The time-dependent terms, though transient, are shown to obey Poynting's theorem. The steady-state part of the Poynting vector is related to the Einstein coefficients. The corresponding electric-energy density is related to the Casimir-Polder potential for a polarizable test body in the field of the source molecules. Similarly the magnetic-energy density is derived and is used in the calculation of energy shifts. Quadratic operators referring to different field points are also discussed.
We give here a field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This approach illustrates however the proper use of the Euler-Lagrange equations and the canonical formalism that fail if one tries to build up the many-body quantum theory starting from the classical theory of point-like particles interacting with electromagnetic fields.
Abstract A canonical transformation is performed on the conventional Hamiltonian for the electromagnetic radiation field and an assemblage of neutral molecules in interaction. The new Hamiltonian has interaction energies expressed in terms of the electromagnetic fields alone and these energies have direct physical significance. The terms linear in e are the multipole interactions, both electric and magnetic, and the term quadratic in e is a generalization of the elementary diamagnetic energy shift. The intermolecular Coulomb energies have cancelled with transverse polarization fields in the new Hamiltonian, although the intramolecular Coulomb potentials are left unaffected. The equations of motion that follow from the new Hamiltonian are deduced. They are the so-called atomic-field equations for the Maxwell fields and Schrödinger equation for an electron wave in a transverse electromagnetic field. The former are the microscopic analogues of Maxwell’s equations in a medium (not restricted to dipole polarization fields) and the latter are dependent on the field strengths alone (not explicit functions of the vector potential).