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American Journal of Physics

On the nature of the Hamiltonian for the interaction of radiation with atoms and molecules: (e/m c)p⋅A, −μ⋅E, and all that

Edwin Albert Power, T. Thirunamachandran

Published Apr 1, 1978• 112 citations
Publisher

Abstract

In nonrelativistic quantum electrodynamics there is a class of equivalent Hamiltonians that describe a system of radiation and atoms or molecules in interaction. Three specific examples, namely, the minimal coupling, the multipolar, and the space-translated forms are analyzed here in detail. Their equivalence is demonstrated. The Lagrangian approach has also been examined to give further insight into the dynamics. The equations of motion are compared. Finally, the process of two-photon absorption by an atom is used as an application of the various interaction Hamiltonians in a discussion of their computational differences.

DOI
10.1119/1.11313

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In nonrelativistic quantum electrodynamics there is a class of equivalent Hamiltonians that describe a system of radiation and atoms or molecules in interaction. Three specific examples, namely, the minimal coupling, the multipolar, and the space-translated forms are analyzed here in detail. Their equivalence is demonstrated. The Lagrangian approach has also been examined to give further insight into the dynamics. The equations of motion are compared. Finally, the process of two-photon absorption by an atom is used as an application of the various interaction Hamiltonians in a discussion of their computational differences.
The equations of motion and the class of Hamiltonians for electrodynamics, both classical and quantal, are considered for the interaction of radiation with bound charges within the electric-dipole approximation. It is emphasized that, despite the uniqueness of the equations of motion for physical variables, the form of the Hamiltonian is not unique and that a physical system is associated with an equivalence class of Hamiltonians. Two of the members of this class are the minimal-coupling and multipolar forms extensively used in quantum optics. It is shown that although different choices of Hamiltonians lead to different first-order equations of motion that connect canonical coordinates and momenta, they lead to the same second-order equations of motion for the physical variables and physical electromagnetic fields. In the quantum theory of electrodynamics the role played by the choice of representation for the operators and for the states is analyzed with care. The choice of representation leads to a freedom additional to that provided by canonical transformations in classical electrodyamics, although the two lead to the same operator equations of motion.
Many dynamical processes of interest are either initiated or probed by light, and their understanding requires some knowledge of this subject. This chapter is included in order to make this text self contained by providing an overview of subjects that are used in various applications later in the text. In particular, it aims to supplement the elementary view of radiation–matter interaction as a time-dependent perturbation in the Hamiltonian, by describing some aspects of the quantum nature of the radiation field. This is done on two levels: The main body of this chapter is an essentially qualitative overview that ends with a treatment of spontaneous emission as an example. The Appendix gives some more details on the mathematical structure of the theory. In elementary treatments of the interaction of atoms and molecules with light, the radiation field is taken as a classical phenomenon.
Considers aspects of the non-relativistic quantum theory of radiation that occur when the coordinates and momenta of all material particles (nuclei as well as electrons) are treated as dynamical variables. The formalism gives the complete scheme of atomic field equations for an ensemble of ions, free electrons, atoms or molecules and comprises the multipolar Hamiltonian for systems composed solely of electrically neutral atoms or molecules. It also deals with effects due specifically to motion of the nuclei, such as the Doppler effect or the recoil that occurs when photons are emitted, scattered or absorbed.
The use of two of the most often employed forms of the Hamiltonian describing the interaction of atoms and molecules with an electromagnetic field is examined using the example of spontaneous emission. It is shown that formally the same initial state corresponds in the cases of different Hamiltonians to physically different initial conditions: a stationary Coulomb field exists in the initial state in one case and a nonstationary field, arising upon sudden excitation of the atom, exists in another. The differences in the initial conditions determine the differences in the photon spectra arising upon subsequent emission. The choice of initial conditions in real situations can be made by making a physical analysis of the excitation process. The Hamiltonian is transformed from one form to another. It is emphasized that the Hamiltonians are equivalent only if the states are correspondingly transformed. The Lamb-Retherford experiment is analyzed qualitatively.
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.