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QM- A bit of manipulation of expectation values.

  1. Apr 18, 2009 #1

    trv

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    1. The problem statement, all variables and given/known data
    The variance of an observable Qhat in a state with wavefunction psi is,

    (delta Qhat)2=<(Qhat-<Qhat>)2>

    Show that this can be written as,

    (delta Qhat)2=<Qhat2>-<Qhat>2

    2. Relevant equations

    As above.

    3. The attempt at a solution

    (delta Qhat)2=<Qhat2-Qhat<Qhat>-<Qhat>Qhat+<Qhat>2>
    L.H.S=<Qhat2-2Qhat<Qhat>+<Qhat>2>
     
  2. jcsd
  3. Apr 18, 2009 #2
    Right, you've got it

    [itex]\langle \hat{Q}^2 -2\hat{Q}\langle \hat{Q} \rangle +\langle \hat{Q} \rangle \rangle[/itex]

    Now recognize that [itex]\langle X \rangle [/itex] is just a number (the expectation value) and thus the expectation value of an expectation value is just the expectation value (that's a mouth full) i.e. [itex]\langle \langle \hat{Q} \rangle \rangle = \langle \hat{Q} \rangle[/itex] and so on. Therefore you get:

    [itex]\langle \hat{Q}^2 \rangle - 2 \langle \hat{Q} \rangle \langle \hat{Q} \rangle + \langle \hat{Q} \rangle ^2 [/itex]

    and I assume you can take it from there
     
  4. Apr 18, 2009 #3

    trv

    User Avatar

    Yep that helps. Thanks Maverick.
     
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