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Harmonic Oscillator Calculations

  • K. T. Hecht
Part of the Graduate Texts in Contemporary Physics book series (GTCP)

Abstract

For many calculations involving 1-D harmonic oscillator wave functions, it is useful to introduce the Bargmann transform through the kernel function
$$ A(k,x)=\frac{1}{{{\pi^{\frac{1}{4}}}}}exp(-\frac{1}{2}{k^2}+\sqrt{2kx}-\frac{1}{2}{x^2}) $$
(1)
, where k is a complex number. Given a square-integrable function, ψ(x), its Bargmann transform, F(k), is given by
$$ F(k)=\int_{-\infty}^\infty{dx\psi (x)A(k,x)} $$
(2)
, where
$$ \psi(x)=\frac{1}{\pi}\int{{d^2}}k{e^{-kk*}}A(k*,x)F(k) $$
(3)
and the integral is over the 2-D complex k-plane; i.e., wiht k=a+ib,
$$ \int{{d^2}}k\bar= \int_{-\infty}^\infty{da\int_{-\infty}^\infty{db}} $$
(4)
.

Keywords

Harmonic Oscillator Hermite Polynomial Completeness Relation Harmonic Oscillator Potential Nonzero Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • K. T. Hecht
    • 1
  1. 1.Department of PhysicsUniversity of MichiganAnn ArborUSA

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