Expected value of Q(x,p)

  1. nicksauce

    nicksauce 1,275
    Science Advisor
    Homework Helper

    To find the expected value of Q(x,p) we evaluate [itex]<\psi|Q(x,-i\hbar \frac{\partial }{\partial x})|\psi>[/itex]. But what do you do if you want to find, say, <p^(3/2)>. How do you raise the derivative operator to the three-halves?
  2. jcsd
  3. Re: <p^(3/2)>

    Expand the state in momentum eigenstates.
  4. Re: <p^(3/2)>

    You would make like you do for angular momentum. Since you can't take square roots of operator, you have to study the square of the quantity you are interested in. So in your case, you would have to settle for [tex]\sqrt{ \langle p^3\rangle}[/tex], but you would have to put absolute values somewhere.

    ...edit: go with count iblis' suggestion instead
  5. clem

    clem 1,276
    Science Advisor

    Re: <p^(3/2)>

    Fourier transform the wave function to the momentum representation, getting
    [tex]\phi(p)[/tex]. Then p is just like a c number.
  6. nicksauce

    nicksauce 1,275
    Science Advisor
    Homework Helper

    Re: <p^(3/2)>

    Okay, thanks for the replies.
  7. Re: <p^(3/2)>

    Next problem:

    How would you handle:

  8. Re: <p^(3/2)>

    Formally if f(x) is a nice function you could always calculate the n'th derivative using Fourier transforms, like


    also when [tex]n=\sqrt{pi}/i^5.8[/tex], for example. So the <x^(1/2)p^(3/2)> involves a tripple integral.
  9. clem

    clem 1,276
    Science Advisor

    Re: <p^(3/2)>

    Fourier transform [tex]\psi(x)[/tex] and [tex]x^{1/2}\psi(x)[/tex] separately.
  10. Re: <p^(3/2)>

    Ok, that was too easy for you two. :smile: Let me think of a more difficult problem. Well, why not just consider <f(x,p)> where f is some arbitrary function, like e.g.:

    [tex]f(p,x)=\sqrt{x^2 + p^2}[/tex]

    Now, you can just modify the Fourier transform method and write this too as a triple integral. However, that may not be the simplest way, particularly not in the case when f is given as above. :smile:
  11. clem

    clem 1,276
    Science Advisor

    Re: <p^(3/2)>

    For that you may need a Taylor expansion.
  12. Re: <p^(3/2)>

    Diagonalizing the operator x^2 + p^2 is easier. The eigenstates are the harmonic oscillator eigenstates, let's denote them by |n>. You can thus compute the average as:

    <psi|sqrt(x^2 + P^2)|psi> =

    sum over n of <psi|sqrt(x^2 + p^2)|n><n|psi>

    sqrt(x^2 + p^2)|n> = sqrt[(n+1/2)C]

    were C follows from the usual H.O. algebra (i'm too lazy to compute it right now) So, the average is:

    sum over n of sqrt[(n+1/2)C] |<n|psi>|^2
  13. clem

    clem 1,276
    Science Advisor

    Re: <p^(3/2)>

    Expanding in SHO states may or may not be easier than Taylor expansion.
    This would depend on the original wave function and operator.
    SHO works for the particular operator x^2+p^2, but TE works for most operators.
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?