Delta-potential scattering problem

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The discussion centers on a 1-D quantum system featuring a delta-potential barrier at x = 0 and an infinitely high wall at x = -a, with the system open on the right side. The potential is defined as V = κδ(x) at the barrier, V = ∞ at the wall, and V = 0 elsewhere, leading to a Schrödinger equation that applies in both regions. The wavefunctions in the two regions are expressed as combinations of exponential functions, and boundary conditions are established at the wall and the delta barrier. The user seeks guidance on solving for the constants A, B, C, and D to determine the S-matrix components, noting that previous approaches for fully open systems do not apply here due to the presence of the wall. Clarification is provided that while waves can be reflected between the wall and the delta potential, no waves propagate from the right side of the delta barrier.
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I am looking at a 1-d quantum system with a delta-potential barrier in the centre (at x = 0) and an infinitely high wall on one side of this barrier (at x = -a), while the system is open on the other side.

So the potential V is equal to:
V = κ\delta(x) at x = 0, κ being some constant and δ being the Dirac delta-function
V = ∞ at x = -a, this is where the wall is
and V = 0 everywhere else

Splitting the system into region 1 (the bounded part to the left of the delta-barrier, -a < x < 0) and region 2 (the open part to the right of the delta-barrier, x>0), we get V=0 in both regions and the Schrodinger equation will be of the same form:
-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi

or
\frac{d^2\psi}{dx^2} = -k^2\psi
where k = \frac{\sqrt{2mE}}{\hbar}

And so the wavefunction in both regions has form:
\psi_1 = Ae^{ikx} + Be^{-ikx}
\psi_2 = Ce^{ikx} + De^{-ikx}

The boundary condition at -a gives:
Ae^{-ika} + Be^{ika} = 0
Or equivalently:
Acos(ka) = Bsin(ka)

And using the standard trick of integrating the Schrodinger equation on a small integral around the origin [-ε,ε] and taking the limit ε→0, we get another condition:
ik (C-D-A+B)= \frac{2m\kappa}{\hbar^2}(A+B)

which by rearranging we can also express as:
C-D = A(1-2i\beta) - B(1+2i\beta)
where \beta= \frac{m\kappa}{k\hbar^2}

And at this stage I am a bit stuck about what to do next with these boundary conditions, in order to solve for the constants A,B,C,D and get the S-matrix components. I have already solved this same problem in the case where there is a wall on either side of the delta-barrier, and in the case where the system is open on either side, but I'm not sure how to proceed with this semi-open case. Any help, hints or advice would be greatly appreciated, thanks.
 
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What did you do to determine the constants in the case where its open on either side? I'm guessing you can follow a similar procedure here for the open side. That plus normalization should be enough.
 
In that case I simply took the constant D to be 0, since it represents the amplitudes of waves coming in from the right side, and in that example I made the arbitrary assumption to consider waves being propagated from the left only. This made everything a lot easier, but here I don't think I can do the same thing, since I should expect some reflected waves coming back in the opposite direction due to the wall. Am I right in thinking this?
 
That's true in the region between the infinite wall and the delta potential, however, for x>0 you can assume that there are no waves propagating from the right since there's no potential to the right of the delta function were they could be reflected.
 
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