Transmission Coefficient through a delta potential

Homework Statement

Consider an uni-dimensional scattering by a delta function at the origin, given by the potential $V(x) = g \delta(x)$, with $g>0$.

Using the following result, with $G(x)$ being the green function:

$$\Psi (x) = e^{ikx} + g \dfrac{2m}{\hbar}\dfrac{G(x)}{1-2mgG(0)/\hbar^2} = e^{ikx} + g \dfrac{2m}{\hbar 2ik}\dfrac{e^{ikx}}{1-2mgG(0)/\hbar^2}$$

Find the transmission coefficient through the potential.

Homework Equations

$$T = \dfrac{J_t}{J_i}$$

$$J=\dfrac{\hbar}{2mi}(\Psi^\star\dfrac{d\Psi}{dx} - \Psi \dfrac{d \Psi ^\star}{dx})$$

The Attempt at a Solution

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I know I am supposed to use the equations I wrote above, but I don't know what I should consider as the $\Psi_i$ and $\Psi_t$. Any help in that regard?

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vela
Staff Emeritus
I forgot to mention that $G(x) = \dfrac{1}{2ik}e^{ikx}$
You're correct, $i$ means incident and $t$ means transmitted. I don't know how to answer your other queries though. I do not know what's the physical situation in this problem, but I suppose you're right in saying that a wave is coming from one side and gets transmitted and reflected. Again, there's no info about the physical situation, but I'm gonna say that the wave comes from the left ($x<0$) and gets transmitted ($x>0$).