# Transmission amplitude using path-integrals

gulsen
Hello,

As a path-integral newbie, I've been trying to calculate the amplitude for an electron which enters a box (potential within the box is given) at a point to emerge the other edge of the box (it doesn't matter when it exits). For simplicity, I first tried to work out the problem in one dimension, and in discrete space-time. To simplify it even further, I tried with a constant (but non-zero) potential.

I worked out the kernel $$K(b,a)$$, but it -naturally- depends on time spent in the "box". But I don't care when will exit, I care only whether if it can or can not penetrate through.

I have $$\psi(b,t_b) = \int K(b,a) \psi(a,t_a) da$$, and transmission amplitude at $$t_b$$ would be "inner product" of wavefunctions at $$t_b$$ and $$t_a$$ (but well, how? They don't have a variable in common at all! Do I get to expand the wavefunction in eigenstates of position?). So I guess, to get the total amplitiude, I get to compute the amplitude for all times after $$t_a$$, and add them all. Sounds plausible to me, but how would I do an inner product with $$\psi(b,t_b)$$ and $$\psi(a,t_a)$$? Or am I quite off?

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