Transmission amplitude using path-integrals

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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 [tex]K(b,a)[/tex], 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 [tex]\psi(b,t_b) = \int K(b,a) \psi(a,t_a) da[/tex], and transmission amplitude at [tex]t_b[/tex] would be "inner product" of wavefunctions at [tex]t_b[/tex] and [tex]t_a[/tex] (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 [tex]t_a[/tex], and add them all. Sounds plausible to me, but how would I do an inner product with [tex]\psi(b,t_b)[/tex] and [tex]\psi(a,t_a)[/tex]? Or am I quite off?
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