QuantumMechanically Correct Form of Hamiltonian Function for Conservative Systems
Abstract
Dirac showed that, if in the Hamiltonian H momenta η_{r} conjugate to the coordinates ξ_{r} are replaced by (h2πi)∂∂ξ_{r}, the Schrödinger equation appropriate to the coordinate system ξ_{r} is (HE)ψ_{ξ}=0. Applied to coordinate systems other than cartesian this usually leads to incorrect results. The difficulty is here traced partially to the way in which ψ_{ξ} is normalized and partly to the choice of H. In H expressions such as qpq^{1}p and p^{2} are not equivalent, and the simplified form is generally incorrect. A formula satisfying all the requirements of quantum mechanics for a Hamiltonian of a conservative system, in an arbitrary coordinate system, is therefore developed H=12μr=1r=ns=1s=ng^{ 14}p_{r}g^{12}g^{rs}p_{s}g^{ 14}+U This formula is applied to a case of plane polar coordinates and leads to correct results.
 Publication:

Physical Review
 Pub Date:
 November 1928
 DOI:
 10.1103/PhysRev.32.812
 Bibcode:
 1928PhRv...32..812P