Generalization of Quantum Mechanics

The possibility of generalizing quantum mechanics in such a way as to retain its predictive results, while comprehending additional solutions, is examined. It is found that this can be done through a perfected formal correspondence with Hamilton-Jacobi mechanics, by which one is led to consider generalizations of the Heisenberg postulate of the form $p_k{q}_j-q_j{p}_k=S{\delta }_{\mathrm{jk}}$, where $S$ is a quantum analog of Hamilton's principal function. The formalism is shown to be equivalent to a simple change in Hamiltonian, with transformed momentum operators satisfying conventional commutation relations, and with an additional relationship involving formal analogs of the classical "initial constants" adjoined. A particular choice of $S(=\frac{\hslash }{i})$ leads to a theory identical with wave mechanics a part from a constant (unobservable) phase factor on the wave function. The fact that $S$ may possess other, nonconstant values, demonstrated by a specific example, suggests the ability of the mechanical equations to describe a broader class of physical states than has hitherto been investigated.