Derivation of the Bohr Sommerfeld Quantum Conditions From An Asymptotic Solution
Derivation of the Bohr Sommerfeld Quantum Conditions From An Asymptotic Solution
Joseph B Keller
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T, ii) denotes the difference between any two values of S, pt q, t, and n is an integer. Equation (15) may be celled the K r ■ quantum condition'. - 8 - If \}/ denotes a steady state of energy E, then S, is £iven by (12) and V = \ i^ given by (1^). The difference/^ S (q . T^ti) is then given by a line integral over some closed curve starting end ending at a, and (15) becomes (16) / Y2 vs- = ^ ^ • r In fact (l6) must hold for every closed curve in the a sppce since only then will exp(iii'* S, ) ...pnd thus \|; be single-valued at every point. Equation (16) is exsctly the quantum condition postulated by Einstein, Einstein also pointed out that since p = SSj/Sq^i the line integral in (16) hss the same value for ?11 curves which can be deformed into each other without crossing a singularity of the integrand. For example, the integral is zero for a curve which can be deformed into a. [ point. There are usually a finite number of classes of independent curves which cannot be deformed into a point.
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