Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative Systems

Dirac showed that, if in the Hamiltonian H momenta ₑ conjugate to the co-ordinates ₑ are replaced by (h2) {ₑ}, the Schr\"odinger equation appropriate to the coordinate system ₑ is (H-E) _=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 _ is normalized and partly to the choice of H. In H expressions such as qpq^-1p 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=12=nr=1=ns=1g^-1{4}pₑg^1{2}g^rspₒg^-1{4}+U This formula is applied to a case of plane polar coordinates and leads to correct results.