On the transition from classical to quantum mechanics in generalized coordinates

Abstract

The classical Hamiltonian in generalized coordinates is given asH=1/2 Σ _i.k p _i g^ik p _k . We show that there is no operator of the formP _i= −iA(q_i) (∂/∂q_i)+G_i(q_i) (note that the Hermitian momentum operatorP ^H_i is of this form) such that the quantum Hamiltonian operatorH _Q is given asH _Q =1/2 Σ _i,k P _i g ^ik P _k or1/2 Σ _i,k g ^ik P _i P _k , etc. In order to maintain a direct transition of this sort from classical to quantum theory, using the classical Hamiltonian as a starting point, we must rely on our previous prescriptions, writing the quantum Hamiltonian asH _Q =1/2 Σ _i,k P _i ⁺ g ^ik P _k , whereP ⁺_i denotes the adjoint of the operatorP _i=−ih ∂/∂q_i.