Abstract
The classical Hamiltonian in generalized coordinates is given asH=1/2 Σ i.k p i gik p k . We show that there is no operator of the formP i= −iA(qi) (∂/∂qi)+Gi(qi) (note that the Hermitian momentum operatorP Hi 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.
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D. I. Blokhinstev,Quantum Mechanics, D. Reidel Publishing Co., Dordrecht, Holland (1964).
G. R. Gruber,Int. J. Theoret. Phys. 6, 1, 31 (1972).
G. R. Gruber,Int. J. Theoret. Phys. 7, 4, 253 (1973).
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Gruber, G.R. On the transition from classical to quantum mechanics in generalized coordinates. Found Phys 6, 111–113 (1976). https://doi.org/10.1007/BF00708669
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DOI: https://doi.org/10.1007/BF00708669