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Normalizing a Wavefunction

  1. May 15, 2015 #1


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    1. The problem statement, all variables and given/known data
    A particle is described by the wavefunction:
    [tex]\psi (x) = \{ \begin{array}{*{20}{c}}
    {A\cos (\frac{{2\pi x}}{L}){\quad\rm{for }} - \frac{L}{4} \le x \le \frac{L}{4}}\\
    {0{\quad\rm{otherwise }}}

    (a) Determine the normalization constant A
    (b) What is the probability that the particle will be found between x=0 and x=L/8 if a measurement is made?

    2. Relevant equations


    3. The attempt at a solution

    Okay, just verify my logic for me:

    (a) We integrate the wave function from -L/4→L/4, where the particle has a 100% chance of appearing, and set this integral equal to 1 (for the probability just mentioned):

    [tex]1 = \int\limits_{ - \frac{L}{4}}^{\frac{L}{4}} \psi {(x)^*}\psi (x)dx[/tex]

    Integrating this (using Mathematica) and solving for A seems to point at [itex]A = \pm \frac{2}{{\sqrt L }}[/itex].

    I'll only assume the positive value is valid since I'm not sure we can have a negative probability.

    (b) Assuming that the normalization constant in part (a) is correct, we use our new normalization constant in our wavefunction and perform the same integration, this time over 0→L/8. Performing this operation seems to yield 40.9% (0.409).
  2. jcsd
  3. May 15, 2015 #2
    Yes your logic is correct. You're right that we can't have a negative probability, but since the probability comes from ##\psi ^*\psi##, the normalisation constant only ever appears as ##A^2##, so it doesn't matter whether you pick the positive or negative value. Generally we pick the positive one just because positive numbers are easier to think about.
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