Gauges Hamiltonian transformation

Then, two things (that are actually interdependent) happen (1) The field intensity F = 0, (2) There exists a unique gauge g(R) and, since F = 0, any apparent field in the Hamiltonian can be transformed away by introducing a new gauge. If, however, condition (1) does not hold, that is, the electronic Hilbert space is truncated, then F is in general not zero within the tmncated set. In this event, the fields A and F cannot be nullified by a new gauge and the resulting YM field is true and irremovable. 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

Ferraro and coll, used canonical transformation of the Hamiltonian to resolve the average optical rotatory power of a molecule into atomic contributions, based on the acceleration gauge for the electric dipole, and/or the torque formalism [151], This method has been applied to the study of the conformational profile of the optical rotatory poser of hydrogen peroxide and hydrazine [152]. 

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... 

This can be accomplished in two ways (i) via a canonical transformation of the Hamiltonian (7) and its (perturbed) eigenstates, assuming the radiation gauge (2) [24-26],... 

This distinction is largely formal, owing to the substantial identity of the unitary time-dependent transformation (8)-(9) with the gauge transformations of the Hamiltonian and its eigenfunctions [21-22]. However, alterna-... 

In quantum mechanics the unitary transformation (8) leads to different gauges for the Hamiltonian, which are sometimes also referred to as different formalisms. An alternative first-order Hamiltonian can be defined from (8)... 

In principle, a chemical shift calculation represents a perturbation theory, because of the presence of an external field Bz and magnetic moments due to the dipole character of nuclei. Therefore, perturbations to the Hamiltonian and the wave function have to be considered. The next important point is that the origin of the vector potential Az is not fixed due to the relation Bz = rot Az- Any change of the gauge origin Rq should not change any measurable observable. Therefore, a gauge transformation of the wave function 1%) and Hamilton operator h is essential... 

As long as ) differs from 4>) by more than a gauge transformation the state 4> ) represents some kind of excited state of the Hamiltonian with potential Vfi. The renormalized energy es associated with <> ) in the unprimed system is given by... 

Now we can see that the change of the gauge origin corresponds to a unitary transformation of the Hamiltonian. Although the wave function is affected by the unitary transformation, the eigenvalues remain unchanged and the Schrodinger equation is solved exactly. 

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... 

Thus, the generated fnuc,n+ is a unitary transformation of mid,n+ This means that the FC function is always a gauge-including function if its initial function is gauge including. Because this can be extended to a general Hamiltonian and initial... 

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