1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    jtbell

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Normalizing Wave Functions Over Multiple Regions
Loading...