I don't understand how we get from the RHS to the last line.

\begin{eqnarray} \left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\phi_n(x) \phi_n^* (y) e^{-i E_n (t_x - t_y)}}\\ &=& - i \delta(t_x - t_y) \delta(x - y) \end{eqnarray}

I understand that $\sum_n \phi_n(x) \phi_n^*(y) = \delta(x-y)$ But I can't figure out where the. $\sum_n e^{iE_n(t_x-t_y)}$ goes to.

  • 3
    $\begingroup$ Hint: $\delta(t)f(t)=\delta(t)f(0)$. $\endgroup$
    – Qmechanic
    May 11, 2021 at 22:00

1 Answer 1


The trick is: $$ \sum_0^{N-1} e^{-iEn(t_x-t_y)} = N\delta(t_x-t_y) $$

  • $\begingroup$ Erm I don't understand, how did you get that relation? Also if you have an $N$, wouldn't the answer be $-iN \delta(t_x-t_y) \delta(x-y)$ ? $\endgroup$
    – chou
    Apr 1, 2019 at 17:08

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