Two Notes on Quantum Conditions

Selection of coordinates for use in evaluating the Sommerfield quantum integrals.—It is well known that these coordinates need not necessarily satisfy Lagrange's equations, and a general criterion for the choice of such variables is given in the present paper, in which it is shown that all sets of $p$'s and $q$'s which (a) satisfy Hamilton's equations, (b) separate the variables, and (c) are of such a character that the average value of the kinetic energy is equal to that of $\frac{1}{2}\Sigma 1^n{p}_i{\dot{q}}_i$, will give the correct energy levels when employed in the quantum integrals $\int p_{i}d{q}_i=n_{i}h$; ($i=1,2,\cdots ··,n$). This general class of coordinate systems includes as special cases the familiar normalized Schwarzschild angle variables and Lagrangian generalized coordinates which separate the variables.