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Path Integrals Harmonic Oscillator

  1. May 10, 2013 #1
    Hi,

    I am reading through the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs and am having a bit of trouble with problem 3-12. The question is (all Planck constants are the reduced Planck constant and all integrals are from -infinity to infinity):
    The wavefunction for a harmonic oscillator is (at t=0): ψ(x,0) = exp[-[itex]\frac{mω}{2h}[/itex] (x-a)[itex]^{2}[/itex]]
    then, using Eq.(3.42) (ψ(x[itex]_{b}[/itex],t[itex]_{b}[/itex])=∫K(x[itex]_{b}[/itex],t[itex]_{b}[/itex];x[itex]_{c}[/itex],t[itex]_{c}[/itex]) * ψ(x[itex]_{c}[/itex],t[itex]_{c}[/itex]) dx[itex]_{c}[/itex]) and the results of problem 3-8 (the Kernel for a harmonic oscillator K=([itex]\frac{mω}{2πihsin(ωτ)}[/itex])[itex]^{1/2}[/itex] * exp[[itex]\frac{imω}{2hsin(ωτ)}[/itex]((x[itex]_{b}[/itex][itex]^{2}[/itex] + x[itex]_{a}[/itex][itex]^{2}[/itex])cos(ωτ)-2x[itex]_{b}[/itex]x[itex]_{a}[/itex])]
    show that
    ψ(x,τ)=exp[-[itex]\frac{iωτ}{2}[/itex]-[itex]\frac{mω}{2h}[/itex](x[itex]^{2}[/itex]-2axe[itex]^{-iωτ}[/itex]+a[itex]^{2}[/itex]cos(ωτ)e[itex]^{-iωτ}[/itex])]

    Now so far as I can tell, a solution is to multiply the wavefunction at time 0 by the kernel from x at t=0 to x at t=τ and then integrate over all x, however (unless I've made an algebra mistake which is always possible) this doesn't give the required answer. Any ideas?

    Thanks,
    Will
     
  2. jcsd
  3. May 10, 2013 #2

    fzero

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    Yes, the evolution of the wavefunction is given by

    $$ \psi(x,t) = \int_{-\infty}^\infty K(x,t;x',0) \psi(x',0) dx'.$$

    This integral is of the form

    $$ \begin{split}
    \int_{-\infty}^\infty \exp{\left[ -A (x')^2 + B x' + C\right]} dx' &= \int_{-\infty}^\infty\exp{\left[ -A \left(x'-\frac{B}{2A}\right)^2 + C + \frac{B^2}{4A} \right]} dx'\\
    & = \sqrt{\frac{\pi}{A}} \exp{\left[ C + \frac{B^2}{4A} \right]}. \end{split}$$

    Hopefully some of this is familiar and you can figure out the algebra.
     
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