Minimal-coupling Hamiltonian

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

In the EDA there is no dependence on the spatial coordinates of fhe elec-fric field or of fhe corresponding vector potential. Specifically, suppose we consider fhe minimal coupling Hamiltonian [2,106] which is obfained by fhe sfandard subsfifution of p by p - A(r), where A r) is fhe vecfor pofenfial. (Bold letters symbolize operators.) This substitution produces the interaction between the atomic electrons and the EMF as [2,106]... 

In a different gauge, it is possible to construct the multipolar Hamiltonian which is obtained by applying a unitary transformation to the minimal coupling Hamiltonian [75-77,106]. In the multipolar Hamiltonian, it is the transverse electric field, and the magnetic field, B(r) (satisfying Maxwell s equation, V x Et = - f), that appear, rather than the vector potential. Now, the interaction is written as... 

The first teirni in Eq. (7) contains the factor (see (5)), i.e. the inverse of the difference factor in ( t), and is therefore independent of whether the minimal-coupling or the multipolar Hamiltonian is used. The second teimi represents a velocity effect on emission with q ui. Thi effect is rather curious since such emission can be perpendicular to V, i.e. its frequency can be Doppler-free. This term can also be obtained directly from the minimal-coupling Hamiltonian, if (ri) is expanded to first order in r -. ... 

In relativistic theory, we apply the minimal coupling recipe to the Dirac Hamiltonian... 

The first two terms are the molecular Hamiltonian and the radiation field Hamiltonian. The molecular Schrodinger equation for the first term in (5.2) is assumed solved, with known eigenvalues and eigenfunctions. Solutions for the second term in (3.4) in vacuo are taken in second-quantized form. Hint can be taken in minimal-coupling form (5.3) allowing for the variation of the radiation field over the extent of the molecule,... 

Assuming the interaction between the oscillator and the field to be described by means of the standard minimal coupling term —(e/mc)pA, we arrive at the following two-dimensional Hamiltonian governing the evolution of the coupled (field oscillator + detector) system ... 

Here // < is the Hamiltonian for the radiation field in vacuo, flmo the field-free Hamiltonian for molecule , and //m( is a term representing molecular interaction with the radiation. It is worth emphasising that the basic simplicity of Eq. (1) specifically results from adoption of the multipolar form of light-matter interaction. This is based on a well-known canonical transformation from the minimal-coupling interaction [17-21]. The procedure results in precise cancellation from the system Hamiltonian of all Coulombic terms, save those intrinsic to the Hamiltonian operators for the component molecules hence no terms involving intermolecular interactions appear in Eq. (1). 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

In order to establish a relativistic hyperfine Hamiltonian operator for a many-electron system one faces the problem of setting up a relativistic many-body Hamiltonian which cannot be written down in closed form. If one considers a one-electron system first one can obtain an exact expression for the hyperfine Hamiltonian starting from the one-electron Dirac equation in minimal coupling to the electromagnetic field ... 

The two conventional Hamiltonians of quantum electrodynamics, namely, the minimal-coupling and the multipolar (i t 2) Hamiltonians are known... 

Within the framework of nonrelativistic quantum electrodynamics, the emission in electric-dipole transitions can be treated using two alternative Hamiltonians for field-matter interaction, i.e. a multipolar Hamiltonian and a minimal-coupling (p ) Hamiltonian, since the two are related by a canonical transformation . In what follows, the results concerning motional effects on the emission will be discussed and checked by showing that they are obtainable from both Hamiltonians. 

This system in which the longitudinal and transverse motions of the emitter are separated to a good approximation, provides a convenient example for the consideration of the motional effects discussed above. As shown by Healy", the canonical transfoimiation from the minimal-coupling to the multipolar Hamiltonian has the same form for any convenient reference-point R (not necessarily the center-of-mass) relative to which the polarizations are defined. This arbitrariness, which amounts to a gauge... 

The components of the 4-potential are given by AT (, A). Note that the vector potential A = A, A, A ) contains the contravariant components of the 4-potential. According to what follows Eq. (5.54), we need to add a Lorentz scalar to the (scalar) Dirac Hamiltonian in order to preserve Lorentz covariance. This Lorentz scalar shall depend on the 4-potential. The simplest choice is a linear dependence on the 4-potential and by multiplication with 7H we obtain the desired Lorentz scalar. Minimal coupling thus means the following substitution for the 4-momentum operator... 

The reason why X can be considered as a simple additive operator is the fact that external vector and scalar potentials hidden in X(f) are simply added to the field-free one-electron Dirac Hamiltonian by the principle of minimal coupling discussed in section 5.4. Therefore, they can easily be separated from the field-free many-electron Hamiltonian Hgi discussed so far. 

The usual way to treat the interaction between electromagnetic fields or nuclear electromagnetic moments and molecules is a semi-classical way, where the fields or nuclear moments are treated classically and the electrons are treated by quantum mechanics. The fields or nuclear moments are thus not part of the system, which is treated quantum mechanically, but they are merely considered to be perturbations that do not respond to the presence of the molecule. They therefore enter the molecular Hamiltonian in terms of external potentials similar to the Coulomb potential due to the charges of the nuclei. This is therefore called the minimal coupling approach. 

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