Solving Dirac Delta Potential: Reflection & Transmission Coefficients

[kcirick]

Question:
Consider the motion of a particle of mass m in a 1D potential $V(x) = \lambda \delta (x)$. For $\lambda > 0$ (repulsive potential), obtain the reflection R and transmission T coefficients.

[Hint] Integrate the Schordinger equation from $-\eta$ to $\eta$ i.e.
$$\Psi^{'}(x=\epsilon )-\Psi^{'}(x=-\epsilon )=\frac{2m}{\hbar^{2}}\lambda\int^{\epsilon}_{-\epsilon}\delta (x)\Psi (x)dx = \frac{2m}{\hbar^{2}}\lambda\Psi (x > 0)$$

What I have so far:
Inside the barrier, the wave function is:

$$\psi (x)= Ae^{\kappa x}+Be^{-\kappa x}$$

where:

$$\kappa = \sqrt{\frac{2m}{\hbar^{2}}\left(V-E\right)}$$

Outside we have wave function in the form of:

$$\psi (x) = Ce^{ikx}+De^{-ikx} x < 0$$
$$\psi (x) = Ee^{ikx} x > a$$

and $R = \frac{|D|^2}{|C|^2}$ and $T = \frac{|E|^2}{|C|^2}$.

I have in my notes how to get the ratio $\frac{D}{C}$ and $\frac{E}{C}$, but how does the hint that was given to me used for? where does the delta function come in play?

I don't really get the hint itself either. How does integrating Schrodinger Equation give me that relation in the hint? I am very lost...

 

[George Jones]

Inside the barrier

There is no "inside the barrier," since a delta function is a width-zero barrier.

1) Write down Schrodinger's equation.

2) Integrate it term-by-term over the interval $(-\epsilon, \epsilon)$.

3) Take the limit as $\epsilon \rightarrow 0$.

Outside we have wave function in the form of:

$$\psi (x) = Ce^{ikx}+De^{-ikx} x < 0$$
$$\psi (x) = Ee^{ikx} x > 0$$

Assume $\psi$ is continuous at $x = 0$. This gives you a relationship between the three coefficients.