# The relation between the action of tunneling and the energy

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.

Thanks,

• 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. – Trimok Nov 14 '13 at 12:26