Electric and magnetic interactions. The classical approach

So far we have been concerned solely with the Schrodinger equation for the electrons in an isolated molecule with fixed nuclei, although we know (Section 1.1) that this equation is sometimes inadequate to describe the phenomena in which we are interested. In particular, we may need to consider the interaction of a molecule with an external electromagnetic field and also the internal electromagnetic interactions arising from spins and orbital motion. 

To treat these effects properly, it is necessary to employ field theory and relativistic quantum mechanics. However, it is instructive at this stage to consider the incorporation in a purely classical way of an external field. Since some of the fields of interest are time-dependent, we shall start with the time-dependent Schrodinger equation for a collection of particles with charges q/ and masses /n,t namely 

From classical electromagnetic theory, it is known that the electric field strength E and the magnetic flux density B may be derived from a scalar 

This solution however, is not unique, for two different choices of origin would give two alternative values of A at any given point in space, though the held itself must be independent of origin. More generally, replacement of the potentials by new potentials according to 

To set up the Hamiltonian, we must return to the axiomatic foundations of quantum mechanics and find a set of canonically conjugate momenta and coordinates in the classical problem, in order to set up a quantum-mechanical Hamiltonian operator. To find the classical momenta and coordinates, we must find the Lagrangian L appropriate to charged particles moving in the field. The momentum conjugate to is then 

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