In the semi-classical physics, the probability of the penetration through a barrier is given by $$ p \sim \exp \left( - A_{0} (E) \right), $$ where $A_0$ is the imaginary part of the action and $E$ is the energy. Now, my problem is that I do not understand the following relation: $$ \frac{\partial A_0}{\partial E}= -2 \tau_0 $$ where $\tau_0$ is the imaginary time of the motion under the barrier between two points.

I would appreciate if someone could explain why the last relation holds.


  • 1
    $\begingroup$ Very rough : with $\psi = e^{- E\tau} = e^{- S(E)}$, you could write $\tau \psi= -\frac {\partial S(E)}{\partial E} \psi$. Generalizing $\tau = -\frac {\partial S(E)}{\partial E}$ to arbitrary $S(E)$, and noting that $p = \psi^2$, so that $A_0(E) = 2 S(E)$, you get your relation. $\endgroup$ – Trimok Nov 14 '13 at 12:26

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