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Homework Help: Expectation value and momentum for an infinite square well

  1. May 6, 2016 #1
    1. The problem statement, all variables and given/known data√
    A particle in an infinite square well has the initial wave function:

    [itex]Ψ(x, 0) = A x ( a - x )[/itex]

    a) Normalize Ψ(x, 0)

    b) Compute <x>, <p>, and <H> at t = 0. (Note: you cannot get <p> by differentiating <x> because you only know <x> at one instance of time)

    2. Relevant equations

    3. The attempt at a solution
    For part a, I figured out A = [itex]\sqrt{30 / a^5}[/itex]

    I'm sort of confused for part b. For <x>, I set up the integral like this:

    [itex]\int_{0}^{a}x Ψ(x, 0)^2 dx[/itex]

    And got [itex]\frac{a^6 A^2}{60}[/itex], but I'm not sure if I got it right.

    For <p> and <H>, I don't know how to set up these integrals. How would I set up these integrals? I don't need you to solve it for me, I just wanted to know how to set them up.

    Thank you for reading and helping.
  2. jcsd
  3. May 7, 2016 #2


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    How does the momentum operator look like in position space, i.e. ##p_x\psi(x) =\ldots##?
  4. May 7, 2016 #3


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    If you plug in your result for ##A##, you get ##\langle x \rangle = \frac a2##. Given the symmetry of ##\Psi(x,0)##, does the answer seem reasonable?
  5. May 7, 2016 #4
    I figured it out and got 0 for <p>. That seems reasonable right?

    Ah I didn't think of that. It makes sense.
  6. May 7, 2016 #5
    I also figured out <H> which turns out to be ##\frac{5 h^2}{m a^2}## which seems pretty reasonable too right?
  7. May 7, 2016 #6


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