Delta-potential scattering problem

[fast_eddie]

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.

 

[CFede]

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.

 

[fast_eddie]

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?

 

[CFede]

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.

 

[paranormal]

The Delta-potential scattering problem is a common and interesting problem in quantum mechanics. It involves a one-dimensional system with a delta-potential barrier and an infinitely high wall on one side. The potential in this system is described by V = κ\delta(x) at x = 0, where κ is a constant and δ is the Dirac delta-function. This potential creates a barrier at x = 0, while the system is open on the other side.

To solve this problem, we can split the system into two regions: 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). In both regions, the Schrodinger equation will have the same form, with V = 0. This leads to the wavefunction having the form \psi_1 = Ae^{ikx} + Be^{-ikx} and \psi_2 = Ce^{ikx} + De^{-ikx}, where k = \frac{\sqrt{2mE}}{\hbar}.

The boundary condition at x = -a gives us Acos(ka) = Bsin(ka), and using the standard trick of integrating the Schrodinger equation on a small interval around the origin and taking the limit as the interval goes to zero, we get another condition: ik(C-D-A+B) = \frac{2m\kappa}{\hbar^2}(A+B). This can also be expressed as C-D = A(1-2i\beta) - B(1+2i\beta), where \beta = \frac{m\kappa}{k\hbar^2}.

To solve for the constants A,B,C,D and get the S-matrix components, we need to use these boundary conditions and solve for the unknowns. This can be a challenging task, but there are several methods that can be used, such as the boundary-matching method or the transfer matrix method. In this semi-open case, we need to take into account that the system is only open on one side, which will affect the boundary conditions and the solution.

I would suggest consulting with a textbook or a more experienced colleague for guidance on how to proceed with solving this problem. It may also be helpful to look at other similar problems and their solutions for inspiration. Good luck