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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

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# Path Integrals Harmonic Oscillator

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