Generalizing the exact quantization rule to multiple real turning points

TL;DR

This paper extends the exact quantization rule to systems with multiple real turning points, demonstrating its stability and accuracy through applications to specific potentials.

Contribution

It generalizes the exact quantization rule to cases with multiple real, even-numbered turning points, enhancing its applicability.

Findings

Wave functions are stable between adjacent turning points.

Quantization condition yields integer multiples of π.

Validated with double square well and biharmonic potentials.

Abstract

We generalize the exact quantization rule to multiple turning points, which are all on the real axis and are even in number. We found that when we take wave functions of different energy levels, they are stable between two adjacent turning points and are always integer multiples of {\pi}. We verify the effectiveness of the exact quantization rule by examining double square well potential and biharmonic potential.