An Improved Semi-Classical Approximation Based on Heisenberg's Matrix Mechanics

Ching-Teh Li；Abraham Klein

Chinese Journal of Physics

39卷6期（2001 / 12 / 01）

555 - 564

英文

We have previously shown that the WKB and the Einstein-Brillouin-Keller (EBK) semiclassical quantization methods can be derived within a framework provided by Heisenberg matrix mechanics. Based on the relationship between quantum mechanical matrix elements and classical Fourier components, in a form emphasized in our earlier work, we suggest a modification of the semiclassical calculation that yields markedly improved values for the matrix elements of the elementary position and momentum operators, especially for lowlying states where the WKB values are poorest. The computational framework also provides quantum-mechanical sum rules for the energies that yield similarly improved values when evaluated with the new matrix elements. The scheme is illustrated by application to the quartic oscillator.