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Homework Help: Normalizing Wave Functions Over Multiple Regions

  1. Nov 13, 2013 #1
    1. The problem statement, all variables and given/known data

    I need to normalize the following wave function in order to determine the value of the coefficients. This is from the basic finite square well potential.

    [itex]\Psi(x) = Ae^{k_{1}x},for \ x < -a/2[/itex]
    [itex]\Psi(x) = Csin(k_{2}x),for \ -a/2 \leq x \leq a/2[/itex]
    [itex]\Psi(x) = De^{-k_{1}x}, for \ x > a/2[/itex]

    2. Relevant equations

    [itex]\int\left|\Psi(x)\right|^{2} dx = 1[/itex]

    3. The attempt at a solution

    Do I do an integral for each region, with the limits of integration being the boundaries of each region, and that integral normalized to 1 for each of those regions? Or do I add up those integrals with the same limits of integration and then set that equal to 1?
    Last edited: Nov 13, 2013
  2. jcsd
  3. Nov 13, 2013 #2
    You use the fact that the wavefunction is continuous at each connecting point,
    (-a/2), a/2, to write C and D in terms of A.

    Then you integrate the piecewise-function squared over the whole interval to tell
    you what A should be.

    [also this might be a solution to the finite square well but it's not the most general solution, which
    allows cos(kx) in the middle as well. <-- totally meant this to be punny
  4. Nov 14, 2013 #3


    User Avatar

    Staff: Mentor

    This. More explicitly:

    $$\int_{-\infty}^{+\infty} {|\Psi|^2 dx} = 1 \\
    \int_{-\infty}^{-a/2} {|\Psi_1|^2 dx}
    + \int_{-a/2}^{+a/2} {|\Psi_2|^2 dx}
    + \int_{+a/2}^{+\infty} {|\Psi_3|^2 dx} = 1$$

    This is the same thing as if you were to integrate a single function over the entire range from -∞ to +∞, by splitting up that range into three pieces and doing those as separate integrals for whatever reason.
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