Noncommuting limits of oscillator wave functions

Quantum harmonic oscillators with spring constants k > 0 plus constant forces f exhibit rescaled and displaced Hermite—Gaussian wave functions, and discrete, lower bound spectra. We examine their limits when (k, f) → (0, 0) along two different paths. When f → 0 and then k → 0, the contraction is...

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Bibliographic Details
Published inPhysics of atomic nuclei Vol. 70; no. 3; pp. 513 - 519
Main Authors Daboul, J., Pogosyan, G. S., Wolf, K. B.
Format Journal Article
LanguageEnglish
Published New York Springer Nature B.V 01.03.2007
Subjects
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
CONTRACTION
Harmonic functions
HARMONIC OSCILLATORS
HYSTERESIS
Lower bounds
Mathematical problems
MATHEMATICAL SOLUTIONS
OSCILLATORS
PARITY
Plane waves
SPECTRA
SYMMETRY
WAVE FUNCTIONS
WAVE PROPAGATION
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Summary:Quantum harmonic oscillators with spring constants k > 0 plus constant forces f exhibit rescaled and displaced Hermite—Gaussian wave functions, and discrete, lower bound spectra. We examine their limits when (k, f) → (0, 0) along two different paths. When f → 0 and then k → 0, the contraction is standard: the system becomes free with a double continuous, positive spectrum, and the wave functions limit to plane waves of definite parity. On the other hand, when k → 0 first, the contraction path passes through the free-fall system, with a continuous, nondegenerate, unbounded spectrum and displaced Airy wave functions, while parity is lost. The subsequent f → 0 limit of the nonstandard path shows the dc hysteresis phenomenon of noncommuting contractions: the lost parity reappears as an infinitely oscillating superposition of the two limiting solutions that are related by the symmetry.
ISSN:1063-7788
1562-692X
DOI:10.1134/S1063778807030106