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 1/2Σ1ⁿpᵢ ̇ \.qᵢ, will give the correct energy levels when employed in the quantum integrals ∫pᵢdqᵢ=nᵢh; (i=1,2,⋯··,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.Combination of Hamilton's principle with Trkal's method for quantizing conditionally periodic systems is proposed which, like Hamilton's principle, determines the character of the motion permitted by the Newtonian dynamics, and in addition quantizes the orbits by specializing the constants of integration in accordance with the conventional quantum conditions. This variation principle includes formally in a single equation the results of classical dynamics and the Sommerfield quantum conditions, which together determine the sizes and shapes of electronic orbits. The principle is illustrated by an application to a non-linear oscillator in which the potential involves the second and fourth powers of the displacement from the equilibrium position.