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Homework Help: QM binding energy and reflection coefficient

  1. Oct 10, 2008 #1
    1. The problem statement, all variables and given/known data

    consider a potential given by

    V(x) = infinity x < 0
    = 0 x > a
    = a negative function of x in between

    suppose it is known that the interior wave function is such that

    (1/u) (du/dx) at x=a = f(E)

    a. what is the binding energy of a bound state in terms of f(Eb)?

    b. Suppose f(E) is a very slowly varying function of E so that we can take it to be a constant. Calculate the reflected amplitude R(k) in terms of f if the wave function for x > a has the form
    e-ikx + Reikx, and check that

    (the absolute value of R(k))2 = 1

    2. Relavant equations

    k2= 2mE/hbar2

    3. The attempt at a solution

    a. to find the binding energy I calculated f(E) by taking u at x > a to be Teikx so that
    1/u du/dx = ik

    I know that the bound states are discrete solutions where E < 0 and that in order to find them I must match the u at x > a to u in the well at x = a but how do I do this if I don't know the equation for u in the well?

    b. How can the wave function at x > a be e-ikx + Reikx?

    why is it important that f(E) be a constant?

    where do I start?


    Thank you,

    Felicity
     
  2. jcsd
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