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 $<x>_t$?

  • $\begingroup$ Your quantity is the Green's function $G(x,t;{x_0},0)$. $\endgroup$
    – Urgje
    Sep 19, 2015 at 9:23
  • $\begingroup$ Thank you for comment, Do you have more detail or a reference? $\endgroup$
    – Abolfazl
    Sep 20, 2015 at 7:54

1 Answer 1


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.