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Delta potential: Transmission coefficient

  1. Mar 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider a double delta potential given by [itex]V(x) = c_+ \delta (x + \frac{L}{2}) + c_- \delta (x - \frac{L}{2})[/itex]. 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:

    [itex]F = A \cdot \frac{1}{(1 - i\beta_+)(1 - i\beta_-) + \beta_+ \beta_- e^{2ikL}}[/itex]

    With [itex]\beta_{\pm} = \frac{m c_{\pm}}{\hbar^2 k}[/itex] and [itex]k = \frac{\sqrt{2mE}}{\hbar}[/itex].

    1) Calculate the transmission coefficient T if [itex]c_+ = c_- = c[/itex] and the coefficient T' for [itex]c_+ = -c_- = c[/itex]. 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.

    3. The attempt at a solution

    1) I suppose we now have [itex]\beta_+ = \beta_- = \beta[/itex], so:

    [itex]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} [/itex]

    [itex]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 [/itex]

    Before I go further, is this correct?

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

    [itex]T = |\frac{F}{A}|^2 = \frac{1}{(1 - i\beta)^2}[/itex]

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