On the transition from classical to quantum mechanics in generalized coordinates

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(qi) (∂/∂qi)+Gi(qi) (note that the Hermitian momentum operatorP i H 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 ∂/∂qi