Calculating the probability of transmission of a wave function between two delta distributions

Question

Given is the potential :$V(x) = \frac{-\hbar^2}{m}D\delta(x+a) - \frac{-\hbar^2}{m} D\delta(x-a)$ with $a >0$ and $ D > 0$. A stream of particles from the positive $x$-axis are falling towards the potential and I want to calculate the probability of transmission. I would need help with the boundary conditions. My Ansatz is: $\psi_1(x) = Ae^{ikx}+Be^{-ikx}$ for the incoming wave, $\psi_2(x) =Ce^{ikx}+De^{-ikx}$ for the wave between the delta distributions and $\psi_3(x) = Ee^{ikx}+Fe^{-ikx}$ for the wave function $\psi > a$ with $F = 0$. These are my conditions until now: $\psi_1(-a) = \psi_2(-a)$ $\psi_2(a) = \psi_3(a)$ What are the other conditions? I was thinking of calculating the transfer matrix first and afterwards the transmission.

Answer 1

Ok your ansatz about the form of the $\psi$. Now, the other conditions you need come from the discontinuity of the derivative through the delta potential. If you take the time-independent Schrödinger equation and integrate it around a delta (see, for example, here for the steps), then it can be shown that in the case of a potential like $V(x)=\gamma\delta(x-x_0)$ the derivative must possess a discontinuity given by $$ \left.\frac{d\psi}{dx}\right|_{x\to x_0^+} - \left.\frac{d\psi}{dx}\right|_{x\to x_0^-} = \frac{2m\gamma}{\hbar^2}\psi(x_0) $$ in your case $|\gamma|=D\hbar^2/m$ and hence for the two $\delta$s you have $$ \left.\frac{d\psi}{dx}\right|_{x\to -a^+} - \left.\frac{d\psi}{dx}\right|_{x\to -a^-} = -2D\psi(-a) $$ and $$ \left.\frac{d\psi}{dx}\right|_{x\to +a^+} - \left.\frac{d\psi}{dx}\right|_{x\to +a^-} = +2D\psi(+a) $$

Since you have 6 constants $A,B,\dots,F$ to determine, these 2 conditions plus the 2 you wrote plus the $F=0$ will leave us with just one free parameter that should be absorbed in the computation of the reflection and transmission probabilities.

Edit: in the last line I originally wrongly wrote $\psi(-a)$ instead of $\psi(+a)$, now it is corrected.