Minimal electromagnetic coupling

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

Consider now the inclusion of external fields. We do so by the recipe provided by the principle of minimal electromagnetic coupling. The quantized version of the substitutions (114) can be written in covariant notation as... 

Electric and magnetic fields are introduced in terms of their potentials through the principle of minimal electromagnetic coupling [11]... 

The two transducers form an angle of 90° in order to minimize direct coupling between transmitter and receiver. In order to eliminate low-frequency noises due to electromagnetic pickup in cables connecting the receiving... 

For the same reason it is not clear, how to modify the equation for the inclusion of external fields. The principle of minimal coupling p —> p — A, E E + V for the (scalar) square-root Klein-Gordon equation was critizised by J. Sucher [4], who states that there are solutions ip x) and electromagnetic potentials, such that the Lorentz transformed solution is not a solution of the equation with the Lorentz-transformed potentials. Moreover, the nonlocal nature of the equation means that the value of the potential at some point influences the wave function at other points and it is not clear at all how one can interpret this. 

The Dirac equation, with minimal coupling to an external electromagnetic field a x) can be written... 

It is important to note that minimal coupling requires specification of charge. We are interested in electronic solutions and accordingly choose q = e. The positronic solutions are obtained by charge conjugation. We obtain the relativistic wave equation for the electron in the presence of external electromagnetic fields which we shall write as... 

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 ... 

CL is a well-established spectrometric branch of analytical chemistry based on the production of electromagnetic radiation from a chemical reaction (Campana and Baeyens, 2001 Su et ah, 2007). Due to its minimal instrumentation, no external source, simple optical system, and high sensitivity and selectivity, CL-based detection has become a useful tool in liquid phase coupled with flow injection (FI), sequential injection manifolds, liquid chromatography, and capillary electrophoresis systems in the field of environmental, pharmaceutical, clinical, biomedical, and food analysis (Mervartova et al., 2007 Campana et ah, 2009 Fan et al., 2009 Gracia et al., 2009 Wang et al., 2009). 

Therefore, we will make a series of approximations to this approach. First, we will only use quantum mechanics for the description of the molecule and use classical electrodynamics for the electromagnetic fields. In this semi-classical approach the perturbing fields and nuclear moments are considered to be unaffected by the molecular environment, the so-called minimal coupling approximation. 

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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