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Classical oscillator differential equation is replaced by the corresponding(finite time) difference equation. The equation is, then, symmetrized so thatit remains invariant under the change d going to -d, where d is the smallestspan of time. This symmetric equation has solutions, which come in reciprocallyrelated pairs. One member of a pair agrees with the classical solution and theother is an oscillating solution and does not converge to a limit as d goes to0. This solution contributes to oscillator energy a term which is a multiple ofhalf-integers.

Reciprocal Symmetry and Classical Discrete Oscillator Incorporating Half-Integral Energy Levels