Transmission probability of a massive particle

[mnky9800n]

Homework Statement

Consider the potential $$V=V_0 d[\delta(x-a)+\delta(x-a)]$$. Find the transmission probability of the potential for a particle of mass m and wave number $$\sqrt{\frac{2mE}{\hbar^2}}$$ incident on the potential. Discuss the behavior when $$ka~\pi/2$$.

Homework Equations

Schrödinger's Equation
$$H\psi=E\psi$$

The Attempt at a Solution

The solutions to SE for each region are as follows:
region 1: a < x
$$\psi_1 = e^{ikx}+Re^{-ikx}$$
region 2: -a < x < a
$$\psi_2 = Ae^{ikx}+Be^{ikx}$$
region 3: a < x
$$\psi_3 = Te^{ikx}$$

Because the $$\delta$$-function means the wave function is continuous but the derivative is not at x = a we can say:

$$\psi_1(-a) = \psi_2(-a)$$
$$\psi_2(a) = \psi_3(a)$$

therefore:

$$1+Re^{2ika} = A+Be^{2ika}$$
$$A+Be^{-2ika} = T$$

This is where I get stuck. I have no idea where to go from here.

 

[vela]

You have 4 unknowns, so you need two more equations to be able to solve for them. Those equations come from considering the derivatives of the wave functions. Integrate the Schrödinger equation between a-ε and a+ε. In the limit as ε→0, you'll get a relationship between the derivatives on the two sides of the delta function. Do the same thing around x=-a. Then you should be in a position to solve for the coefficients.

I'd go into more detail, but I don't have time right now. It's probably done in your book somewhere, or someone else may come by to provide more help if you can't figure it out.