Semiclassical and large quantum number limits of the Schrödinger equation

Pages: 288 - 294, Issue Special, July 2003

Abstract
Full Text (161.8K)

Harald Friedrich - Physik-Department, Technische Universität München, 85747 Garching, Germany (email)

Abstract:
For bound one-dimensional systems, the semiclassical limit ~~h~~ $\to 0$ of the Schrödinger equation generally corresponds to the limit of infinite quantum numbers, and conventional WKB quantization becomes increasingly accurate in this limit. A potential well with a sufficiently strong attractive inverse-square tail supports an infinite dipole series of bound states, but the limit of infinite quantum numbers is not the semiclassical limit in this case. Semiclassical eigenvalues derived via conventional WKB quantization tend to a constant relative error in the large-quantum-number limit when the Langer modification is used. Without the Langer modification the relative error grows exponentially in the limit of large quantum numbers.

Keywords: semiclassical limit, large quantum numbers, inverse-square potentials, WKB quantization, quantum-classical correspondence.

Mathematics Subject Classification: 81Q20.

Received: August 2002;
Revised:
March 2003;
Published: April 2003.