Harmonic Oscillator Calculations

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


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


Harmonic Oscillator Hermite Polynomial Completeness Relation Harmonic Oscillator Potential Nonzero Matrix 
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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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