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 Schrödinger 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 Schrödinger'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.