General Gauge Independence of Diamagnetism plus Paramagnetism

With the use of the generalised multipolar gauge, a completely general proof is obtained for a result asserted by J. H. Van Vleck in 1932 but never fully proved by him. Specifically it is demonstrated that the matrix elements of the magnetic moment operator are independent of the origin of the vector potential for electromagnetic fields that are non-uniform in space and non-constant in time. A general survey of the principles of the quantum mechanics of solid state and molecular magnetism is also given to set the problem in its context. Based on the multipolar gauge, a simple conversion of the interaction Hamiltonian from the form of A.p to E.r plus higher order terms is demonstrated.