I need to find energies of a particle for which the reflection coefficient from the system $U(x)=\alpha(\delta(x)+\delta(x-a))$ will be maximum. I am struggling with this task for 2 days. Here is my try:

Particle is coming from the left. We know that the reflection coefficient is $R=\frac{1}{|B_1|^2}$ where $B_1$ is the amplitude of the reflected wave. The energy spectrum of the particle is continuous (maybe not?). So the only thing we need to find is $B_1$.

We have 3 areas:

  • area 1 for $-\infty<x<0$

  • area 2 for $0<x<a$

  • and area 3 for $x>0$

The solutions of the stationary Sh. Eq. for these areas are ($k=\frac{\sqrt{2\mu E}}{\hbar}, \mu$-is the mass of the particle):

  • $\psi_1=e^{ikx}+B_1e^{ikx}$, we assume that $A_1=1$

  • $\psi_2=A_2e^{ikx}+B_2e^{ikx}$

  • $\psi_3=A_3e^{ikx}$, as there is no reflected wave

We also have 4 boundary conditions:

  1. $\psi_1(0)=\psi_2(0)$
  2. $\psi_2(a)=\psi_3(a)$
  3. $-\frac{\hbar^2}{2\mu}\int_{-\epsilon}^{+\epsilon}\frac{\partial^2\psi}{\partial x^2}dx+\alpha \int_{-\epsilon}^{+\epsilon}(\delta(x)+\delta(x-a))\psi dx=E\int_{-\epsilon}^{+\epsilon}\psi dx$
  4. $-\frac{\hbar^2}{2\mu}\int_{a-\epsilon}^{a+\epsilon}\frac{\partial^2\psi}{\partial x^2}dx+\alpha \int_{a-\epsilon}^{a+\epsilon}(\delta(x)+\delta(x-a))\psi dx=E\int_{a-\epsilon}^{a+\epsilon}\psi dx$

Condition 3 and 4 gives us the behavior of the $\psi$-function near the delta-potentials.

From 1 condition we get: $1+B_1=A_2+B_2$

From 2 condition we get: $A_2e^{ika}+B_2e^{-ika}=A_3e^{ika}$

From 3 condition we get: $-\frac{ik\hbar^2}{2\mu}(1-B_1-A_2+B_2)+\alpha (1+B_1)=0$

From 4 condition we get: $-\frac{ik\hbar^2}{2\mu}(A_2e^{ika}-B_2e^{-ika}-A_3e^{ika})+\alpha (A_2e^{ika}+B_2e^{-ika})=0$

This is the place where I dont know what to do to get the answer with minimum actions. I got one complex transcendental equation for $k$ and also got some complex and huge equation for $B_1$, I am almost sure that they are not correct. Can somebody show me the best way to get to the answer?


closed as off-topic by AccidentalFourierTransform, Jon Custer, heather, user108787, user36790 Nov 20 '16 at 18:43

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – AccidentalFourierTransform, Jon Custer, heather, Community, Community
If this question can be reworded to fit the rules in the help center, please edit the question.


I left that link. In it is solved a very similar double delta potential step by step.


  • $\begingroup$ Please remember that, Manuel has not yet achieved the rep of 50 required to leave comments, (AFAIK) and that he is not answering a homework question, in line with site policy, instead he is pointing the OP towards a resource . ( that the OP could have found easily, I have to say) $\endgroup$ – user108787 Nov 20 '16 at 17:12
  • 1
    $\begingroup$ Sorry, I thought I could be taken as an answer. I will take in account for the next time. $\endgroup$ – user326159 Nov 20 '16 at 17:28
  • $\begingroup$ No, I think you would need to expand on it a bit for an answer, I found the same link as you, when I came back to post it, you were there ahead of me:). If you can put as a comment instead , I would do so though. Link answers are frowned upon, sorry. $\endgroup$ – user108787 Nov 20 '16 at 17:32

Not the answer you're looking for? Browse other questions tagged or ask your own question.