Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative Systems

Dirac showed that, if in the Hamiltonian H momenta η_r conjugate to the co-ordinates ξ_r are replaced by (h / 2πi)∂ / ∂ξ_r, the Schrödinger equation appropriate to the coordinate system ξ_r is (H-E)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^-1p and p² 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=1 / 2μΣr=1r=nΣs=1s=ng^{-1 / 4}p_rg^{1 / 2}g^rsp_sg^{-1 / 4}+U This formula is applied to a case of plane polar coordinates and leads to correct results.