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Quantum mechanics problem

  1. Oct 19, 2014 #1
    1. The problem statement, all variables and given/known data


    Hi, I have a problem I have been trying to do for a few days and I am not getting it. Any hints would be greatly appreciated. The question is from "The physics of Quantum Mechanics" by Binney and Skinner.

    The question states:
    Let ##\psi##(x) be a properly normalised wavefunction and Q an operator on wavefunctions. Let {qr} be the spectrum of Q and let {Ur(x)} be the corresponding correctly normalised eigenfunctions. Write down an expression for the probability that a measure of Q will yeild the value qr.

    Show that ##\Sigma_r P(q_r |\psi) = 1##.

    Show further that the expectation of Q is ## \langle Q \rangle = \int _{-\infty} ^\infty \psi^* Q \psi dx## .

    2. Relevant equations and attempt

    So for the first part, the probability amplitude of measuring qr given the system is in the state ## |\psi\rangle ## is given by ## \langle q _ r | \psi \rangle = \int _{-\infty} ^\infty u_r^*(x) \psi(x) dx## .

    and taking the mod squared of this gives the probability the question is asking for.

    The next part says that summing these probabilities over all r = 1??? I understand what this means (probability of finding a value of q within the spectrum given = 1), but don't know how to show this.

    As for the last part, the expectation value is the sum of the
    probabilities of getting each value of q multiplied by the value qr, so $$ \langle Q \rangle = \Sigma _ r q_r | \int _ {-\infty}^\infty u_r^*(x) \psi(x) dx |^2 $$


    Cant get any further....
     
    Last edited: Oct 19, 2014
  2. jcsd
  3. Oct 19, 2014 #2
    sorry, latex not working let me try again...

    fixed it...
     
    Last edited: Oct 19, 2014
  4. Oct 19, 2014 #3

    vela

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    Try expanding ##\lvert \psi \rangle## in terms of the eigenstates.
     
  5. Oct 20, 2014 #4
    Thanks Vela,

    I am thinking ##|\psi\rangle = \int_{-\infty} ^\infty \psi(x) |x\rangle## But is it also the case that ##\psi(x) = \sum a_r u_r(x)## where the ##a_r##s are probability amplitudes in Q space.

    Can I say that ##|\psi\rangle = \int_{-\infty} ^\infty (\sum a_r u_r(x)) |x\rangle##
     
  6. Oct 20, 2014 #5

    vela

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    Yes, and that would be equivalent to saying ##\lvert \psi \rangle = \sum_r a_r \lvert q_r \rangle##.
     
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