Gauges dipole velocity

Given this approximation, we can transform the Hamiltonian of Eq. (1.44) from the velocity gauge to the so-called length gauge in which the matter-radiation interaction term contains only the dot product of the dipole moment and the electric field. In order to do so we choose x [Eq. (1.5)] as... 

One important step is the choice of the gauge in which to express the two-body relativistic interaction. For the exact solution we know that the result should be gauge independent but this will not hold for approximate solutions (remember the length and velocity forms for dipole radiative transition probabilities). The final result for g(l,2) of Eq. (5) reads ... 

Let us assume that the field is polarized linearly and fhaf Vext(f) is described in the EDA. There is the question of which form to use. It turns out that only in the length form of fhe EDA do fhe expansion coefficients represent directly probability amplitudes, e.g.. Refs. [80, 83, 101]. On the other hand, in the velocity form it is easier to handle numerically the singularity that appears in the free-free dipole mafrix elements [54,105]. Therefore, we choose fhe A(0) p form, in which case, once the solution of the TDSE, 4>(r,f), is obtained, we must multiply it by the phase factor in order for the new coefficients to acquire their correct meaning as probability amplitudes [83, 101]. Furthermore, in order for the initial state at f = 0 to be a state of Ha>j, the "preferential" gauge can be chosen in which A(f) = 0 whenever E(f) becomes zero [[81b], [87]]. 

D.H. Kobe, Gauge-invariant resolution of the controversy over length versus velocity forms of the interaction with electric dipole radiation, Phys. Rev. A 19 (1979) 205. 

In the length gauge, Eq. (2.122), the operator could be the electric dipole or quadrupole operator, defined in Appendix A. It depends on coordinates and momenta of the electrons but it is independent of time, whereas we assume that the time-dependent field. F. (t) does not depend on any electronic variables. The subscript p - again denotes components of a tensor of appropriate rank. On the other hand, in the velocity gauge, Eq. (2.125), the operator is equal to the total canonical... 

It can be shown that the trace of the tensor G, and hence the computed optical rotation of a sample of randomly oriented chiral molecules, is independent of the origin provided that the linear response function satisfies O Eq. 5.46 and that the commutator of O Eq. 5.47 is fulfilled. Consequently, approximate linear response calculations of the length gauge optical rotation depend on the chosen coordinate origin. On the other hand, the trace of the velocity gauge formulation of the electric dipole - magnetic dipole polarizability... 

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