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Showing the expectation values of a system are real quantities

  1. Sep 28, 2013 #1
    1. The problem statement, all variables and given/known data

    A one-dimension system is in a state described by the normalisable wave function Ψ(x,t) i.e. Ψ → 0 for x → ±∞.

    (a) Show that the expectation value of the position ⟨x⟩ is a real quantity. [1]

    (b) Show that the expectation value of the momentum in the x-direction ⟨p⟩ is a real quantity, too. Hint: using integration by parts and normalisability show that ⟨pˆx⟩ = ⟨p⟩∗. [4]


    2. Relevant equations

    1=N∫ψ*ψ.dx

    <x>=∫ψ* xψ.dx over all space

    <p>=∫ψ* -ih dψ/dx.dx over all space

    3. The attempt at a solution

    The difficult aspect of this for me is determining what the correct wave function is. Using the information given I assumed that the correct wave function was e-ax2/2 eiEt/h (where a/2 is an arbitrary constant) as it fits the above requirements (I could be wrong.)

    However, upon normalising and calculating <x> and <p>, the values obtained will of course will be 0 and therefore real as my assumption was a symmetric wave function. This is all well and good, however the question explicitly states to use integration by parts to solve for <p>.

    <p>=N∫(e-ax2/2 eiEt/h) -ih d/dx(e-ax2/2 e-iEt/h).dx over all space

    which gave:

    <p>=N iha∫xe-ax2.dx over all space

    which cannot be integrated by parts as far as I understand-perhaps it can?. Have I got the wrong end of the stick somewhere in my thinking?

    Thanks in advance.
     
  2. jcsd
  3. Sep 28, 2013 #2
    You wave function is incorrect. Do not plug it in until you have done the math. You do not even need a "test" wavefunction for part a.
     
  4. Sep 28, 2013 #3
    Thanks for the prompt reply, but I don't quite follow. Could you clarify what you mean by 'not needing a wave function'? I feel I am approaching the problem from the wrong angle.
     
  5. Oct 1, 2013 #4
    Hmm, perhaps forget what I mentioned previously. You simply need to reconsider your wave function. Perhaps try something of the form [itex] \psi = e^{\pm i(kx-E t/ \hbar)} [/itex]. . .
     
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