American Institute of Mathematical Sciences

Advanced
2003, 2003(Special): 288-294. doi: 10.3934/proc.2003.2003.288

Semiclassical and large quantum number limits of the Schrödinger equation

Harald Friedrich 1,

1. 

Physik-Department, Technische Universität München, 85747 Garching, Germany

Received  August 2002 Revised  March 2003 Published  April 2003

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: inverse-square potentials, quantum-classical correspondence., semiclassical limit, WKB quantization, large quantum numbers.
Mathematics Subject Classification: 81Q2.
Citation: Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288
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