1. Limited time only! Sign up for a free 30min personal 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 a Wavefunction

  1. May 15, 2015 #1

    squelch

    User Avatar
    Gold Member

    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 }}}
    \end{array}[/tex]

    (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

    N/A

    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.
     
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 a Wavefunction
Loading...