An Improved Semi-Classical Approximation Based On Heisenberg'S Matrix Mechanics

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 low-lying 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.