I have a question just like this Phys.SE problem here with a difference that our system is a harmonic oscillator (rather than a free particle). A particle with mass $m$ is connected to a string with spring constant $k$ at $t=0$ with initial state function $\psi(x)=\delta(x-x_0)$. What is the state function at time $t$ and then calculate the $<x>_t$?
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Your quantity is the Green's function $G(x,t;{x_0},0)$. – Urgje Sep 19 '15 at 09:23
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Thank you for comment, Do you have more detail or a reference? – Abolfazl Sep 20 '15 at 07:54
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We have \begin{eqnarray*} \psi (x,t) &=&<x|\exp [iH(t-t_{0})]\psi (t_{0})>=\int dy<x|\exp [iH(t-t_{0})]|y>\psi (y,t_{0}) \\ &=&\int dyG(x,t;y,t_{0})\psi (y,t_{0}) \end{eqnarray*} and in your case $\psi (y,t_{0})=\delta (y-x_{0})$. Here $H$ is the harmonic oscillator Hamiltonian. For the Green's function I could not find an expression on the web.
Urgje
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