Delta potential: Transmission coefficient

[SoggyBottoms]

Homework Statement

Consider a double delta potential given by $V(x) = c_+ \delta (x + \frac{L}{2}) + c_- \delta (x - \frac{L}{2})$. The coherence between the amplitude A of an incoming wave from the left and the amplitude F of the outgoing wave to the right is given by:

$F = A \cdot \frac{1}{(1 - i\beta_+)(1 - i\beta_-) + \beta_+ \beta_- e^{2ikL}}$

With $\beta_{\pm} = \frac{m c_{\pm}}{\hbar^2 k}$ and $k = \frac{\sqrt{2mE}}{\hbar}$.

1) Calculate the transmission coefficient T if $c_+ = c_- = c$ and the coefficient T' for $c_+ = -c_- = c$. Simplify the expressions to show that T and T' are real.

2) Use the coherence equation above to calculate the transmission coefficient T and reflection coefficient R for a single delta potential.

The Attempt at a Solution

1) I suppose we now have $\beta_+ = \beta_- = \beta$, so:

$T = |\frac{F}{A}|^2 = \left(\frac{1}{(1 - i\beta)^2 + \beta^2 e^{2ikL}}\right)^2 = \frac{1}{(1 - i\beta)^4 + 2(1 - i \beta)^2 \beta^2 + \beta^2}$

$T' = |\frac{F}{A}|^2 = \left(\frac{1}{(1 - i\beta)(1 + i \beta) - \beta^2 e^{2ikL}}\right)^2 = \left(\frac{1}{1 + \beta^2 + \beta^2}\right)^2$

Before I go further, is this correct?

2) Since we are dealing with a single delta potential, could I just set for instance $\beta_- = 0$? Then I end up with:

$T = |\frac{F}{A}|^2 = \frac{1}{(1 - i\beta)^2}$

I know it should be $T = \frac{1}{1 + \beta^2}$, so I guess it's not the right approach, but I can't think of anything else.

 

[mark726]

Hello! Your approach for part 1) looks correct so far. To simplify further, you can use the fact that e^{ix} = \cos(x) + i\sin(x) and substitute for \beta_{\pm} and k in the expressions for T and T'. This will help you to show that they are both real numbers.

For part 2), setting \beta_- = 0 would not be the correct approach. Instead, you can use the given coherence equation to calculate the transmission and reflection coefficients for a single delta potential. Remember that for a single delta potential, we only have one coefficient, c, instead of c_+ and c_- as in the double delta potential. So you will need to substitute c for both c_+ and c_- in the coherence equation.