Coulomb gauge condition

While Ho has the familiar form of the sum of the non-relativistic atomic/molecular Hamiltonian, (55), based purely on Coulombic interactions, and the Hamiltonian for free radiation (48), H has the unfamiliar feature of involving the essentially arbitrary Green s function g(x,x ) because no gauge for the vector potential is specified. In particular the form (55) does not require the Coulomb gauge condition. Of course, overall H is gauge-invariant, and observables must be as well, so we need to consider gauge-invariant calculation. 

Each field mode must also satisfy the Coulomb gauge condition, V Ak(r, t) = 0, which, when substituted into Eq. (1.20), implies that... 

As already mentioned, the vector potential can be chosen to satisfy the Coulomb gauge condition... 

This Hamiltonian results from the standard canonical quantization of electrodynamics if it is assumed that particle speeds are negligible compared to the speed of light, and all charge-photon interactions are discarded the Coulomb gauge condition must also be imposed [8],... 

In many cases the response to a uniform magnetic induction field, B, is required, when the vector potential can be written as A = X r, which automatically satisfies the Coulomb gauge condition, V.A = 0. The hamiltonian can then be reduced to the form... 

It is possible to formulate the Coulomb gauge theory in terms of radiation operators which satisfy the subsidiary condition... 

The divergenceless condition of the field A in the Coulomb gauge means that the complex vector a(k) is transverse, so that k a(k) = 0. Then, for every value of k, we can choose an orthonormal trihedron with by the real vectors k/co, ei(k) and e2(k), and we can represent the field a as... 

The electromagnetic field can be represented in a certain gauge which imposes boundary conditions on the potentials. Using the Coulomb gauge with... 

Other convenient specifications for the divergence can be conceived (see below) the gauge defined by equation (8.41) is known as the Coulomb gauge. With use of this gauge condition, equation (8.40) can be rewritten in the form ... 

The advantage of this approach is in the simplicity of both the potential equations and the boundary-value conditions. Biro and Preis (1990) demonstrated that the Coulomb gauge can be enforced by the following boundary-value condition on the surface dV of the modeling region ... 

Due to the Coulomb gauge, the volume integral on the left-hand side of (12.21) is equal to zero. The boundary condition (12.20) provides the same result as well. 

This type of gauge is called the Coulomb gauge. It reflects the condition of independence between the two components of the wave, since A oscillates in directions perpendicular to the wavenumber vector from Eq. (1.74). This corresponds to the introduction of the linearly polarized light as is shown in Table 1.4. 

The vector potential should be fixed somehow and the most frequently used condition is the Coulomb gauge... 

The precise form of this correction depends upon the gauge condition used to describe the electromagnetic field. In the Coulomb gauge, which has been employed more often in relativistic atomic structure, the electron-electron interactions come from one-photon exchange process and is sum of instantaneous Coulomb interaction and the transverse photon interaction. 

Asymptotic Condition.—In Section 11.1, we exhibited the equivalence of the formulation of quantum electrodynamics in the Coulomb and Lorentz gauges in so far as observable quantities were concerned (t.e., scattering amplitudes). We also noted that both of these formulations, when based on a hamiltonian not containing mass renormalization counter terms, suffered from the difficulty that the... 

Then, for both Lorenz and electric gauges (and others such as Coulomb as well), the change from initial (1) to final (2) conditions is characterized by (35). Referring to (4), we can see that this change is characterized by the electric impulse... 

We could find the gauge transformation by applying the Coulomb condition (3.28). But in this case we know that we want to retain just the Coulomb part of the... 

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