# 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?

Last edited:

vela
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