Exact wave functions for generalized harmonic oscillators

Nathan Lanfear; Raquel M. López; Sergei Suslov

doi:10.1007/s10946-011-9223-1

Exact wave functions for generalized harmonic oscillators

Nathan Lanfear
, Raquel M. López
, Sergei Suslov

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time evolution of exact wave functions of generalized harmonic oscillators is determined in terms of the solutions of certain Ermakov and Riccatitype systems. In addition, we show that the classical Arnold transform is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.

Original language	English (US)
Pages (from-to)	352-361
Number of pages	10
Journal	Journal of Russian Laser Research
Volume	32
Issue number	4
DOIs	https://doi.org/10.1007/s10946-011-9223-1
State	Published - Jul 2011

Keywords

Arnold transform
Ehrenfest theorem
Ermakov-type equation and Pinney-type solution
Green's function
dynamic invariant
generalized harmonic oscillators
time-dependent Schrödinger equation

ASJC Scopus subject areas

Atomic and Molecular Physics, and Optics
Engineering (miscellaneous)

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10.1007/s10946-011-9223-1

Cite this

@article{d504d317ebb54317a833c277aaa08f11,

title = "Exact wave functions for generalized harmonic oscillators",

abstract = "We transform the time-dependent Schr{\"o}dinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time evolution of exact wave functions of generalized harmonic oscillators is determined in terms of the solutions of certain Ermakov and Riccatitype systems. In addition, we show that the classical Arnold transform is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.",

keywords = "Arnold transform, Ehrenfest theorem, Ermakov-type equation and Pinney-type solution, Green's function, dynamic invariant, generalized harmonic oscillators, time-dependent Schr{\"o}dinger equation",

author = "Nathan Lanfear and L{\'o}pez, \{Raquel M.\} and Sergei Suslov",

note = "Funding Information: The authors thank Profs. Carlos Castillo-Ch{\'a}vez, Victor V. Dodonov, and Vladimir I. Man{\textquoteright}ko for support, valuable discussions, and encouragement. They are indebted to Dr. Francisco F. L{\'o}pez-Ruiz for kindly pointing out the papers [54, 55] to our attention and for valuable discussions. The authors are grateful to the organizers of the 12th ICSSUR (Foz do Igua¸cu, Brazil, May 02{\textendash}06, 2011) for their hospitality and an opportunity to present the results of this work. Finally, the authors acknowledge the partial support from the National Science Foundation{\textendash}Enhancing the Mathematical Sciences Workforce in the 21st Century (EMSW21), award \# 0838705, the Alfred P. Sloan Foundation{\textendash}Sloan National Pipeline Program in the Mathematical and Statistical Sciences, award \# LTR 05/19/09, and the National Security Agency{\textendash}Mathematical \& Theoretical Biology Institute{\textemdash}Research program for Undergraduates, award \# H98230-09-1-0104.",

year = "2011",

month = jul,

doi = "10.1007/s10946-011-9223-1",

language = "English (US)",

volume = "32",

pages = "352--361",

journal = "Journal of Russian Laser Research",

issn = "1071-2836",

publisher = "Springer New York",

number = "4",

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TY - JOUR

T1 - Exact wave functions for generalized harmonic oscillators

AU - Lanfear, Nathan

AU - López, Raquel M.

AU - Suslov, Sergei

N1 - Funding Information: The authors thank Profs. Carlos Castillo-Chávez, Victor V. Dodonov, and Vladimir I. Man’ko for support, valuable discussions, and encouragement. They are indebted to Dr. Francisco F. López-Ruiz for kindly pointing out the papers [54, 55] to our attention and for valuable discussions. The authors are grateful to the organizers of the 12th ICSSUR (Foz do Igua¸cu, Brazil, May 02–06, 2011) for their hospitality and an opportunity to present the results of this work. Finally, the authors acknowledge the partial support from the National Science Foundation–Enhancing the Mathematical Sciences Workforce in the 21st Century (EMSW21), award # 0838705, the Alfred P. Sloan Foundation–Sloan National Pipeline Program in the Mathematical and Statistical Sciences, award # LTR 05/19/09, and the National Security Agency–Mathematical & Theoretical Biology Institute—Research program for Undergraduates, award # H98230-09-1-0104.

PY - 2011/7

Y1 - 2011/7

N2 - We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time evolution of exact wave functions of generalized harmonic oscillators is determined in terms of the solutions of certain Ermakov and Riccatitype systems. In addition, we show that the classical Arnold transform is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.

AB - We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time evolution of exact wave functions of generalized harmonic oscillators is determined in terms of the solutions of certain Ermakov and Riccatitype systems. In addition, we show that the classical Arnold transform is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.

KW - Arnold transform

KW - Ehrenfest theorem

KW - Ermakov-type equation and Pinney-type solution

KW - Green's function

KW - dynamic invariant

KW - generalized harmonic oscillators

KW - time-dependent Schrödinger equation

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UR - https://www.scopus.com/pages/publications/80053542998#tab=citedBy

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DO - 10.1007/s10946-011-9223-1

M3 - Article

AN - SCOPUS:80053542998

SN - 1071-2836

VL - 32

SP - 352

EP - 361

JO - Journal of Russian Laser Research

JF - Journal of Russian Laser Research

IS - 4

ER -

Exact wave functions for generalized harmonic oscillators

Abstract

Keywords

ASJC Scopus subject areas

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