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Wavefunction boundary condition solve for k

  1. May 10, 2013 #1
    1. The problem statement, all variables and given/known data

    A wave function is given by:
    [itex] \Psi (x) = a cos(2\pi x) + b sin (2\pi x) for\: x<0 \\
    and\\
    \Psi (x) = Ce^{-kx} for\: x>0 \\[/itex]

    Determine the constant k in terms of a, b and c using the boundary conditions and discuss the case a >> b.

    2. Relevant equations

    Wavefunctions and their first order derivatives are continuous at the boundaries. So at x = 0 they will equal each other.

    3. The attempt at a solution

    [itex] acos (2\pi x) + b sin(2\pi x) = Ce^{-kx}[/itex]

    sin 0 = 0 and cos 0 = 1 and exp 0 = 1 therefore;

    [itex] a = C \\

    -(2\pi x)asin(2\pi x) + (2\pi x)bcos(2\pi x) = -ake^{-kx}[/itex]

    again sin 0 = 0 cos 0 = 1 exp 0 = 1 and x = 0 therefore;

    [itex] -ak = 0[/itex]

    Soo pretty sure this is correct so far but not sure on the final step?
     
  2. jcsd
  3. May 10, 2013 #2

    TSny

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    Check your expressions for the derivatives of the sine and cosine functions. Did you use the chain rule properly?
     
  4. May 10, 2013 #3
    Ahh forgot to get rid of the x so that would mean k = 2pi*b/a. And for cases where a<<b then it is just 2pi*b?
     
  5. May 10, 2013 #4

    TSny

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    I get a different sign for k. The initial problem statement says to consider the case a >> b.

    The question seems a little odd to me. The constants a, b, c need not be positive numbers (or even real for that matter.) Anyway, I guess you could make a conclusion about the size of k under the assumption that |a| >> |b|.
     
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