Quantization scheme for arbitrary one-dimensional potential wells

A formalism that utilizes the analytic transfer matrix technique is applied to the Schrödinger equation. This approach leads to proofs that the phase loss at a turning point is exactly equal to π. We also show the existence of the phase contributions devoted by the scattered subwaves, which to our knowledge, have never been taken into account in previous works. Subsequently, an exact quantization condition, which differs essentially from the WKB method, is introduced for arbitrary potential wells.