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I Factorising expectation values

  1. May 24, 2017 #1

    dyn

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    Hi.
    I came across the following in the solution to a question I was looking , regarding expectation values of momentum in 3-D
    < p12p22p32 > = < p12 > < p22 > <p32 >
    ie. the expectation value has been factorised. I can't figure out why this is true and also why it doesn't apply to the following. eg. < p12 > ≠ < p1 > < p1 > ?
    Any thoughts would be appreciated. Thanks
     
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  3. May 25, 2017 #2

    vanhees71

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    It's only true if the different momentum components are uncorrelated, i.e., if the probability distribution factorizes;
    $$P(\vec{p})=P_1(p_1) P_2(p_2) P_3(p_3).$$
    Then it's very easy to see that the first equation is correct.

    Of course, this does not hold for ##\langle p_1^2 \rangle##, because obviously
    $$\langle p_1^2 \rangle=\int_{\mathbb{R}} \mathrm{d} p_1 p_1^2 P(p_1),$$
    while
    $$\langle p_1 \rangle^2=\left [\int_{\mathbb{R}} \mathrm{d} p_1 p_1 P(p_1) \right]^2 \neq \langle p_1^2 \rangle.$$
     
  4. May 25, 2017 #3

    dyn

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    Thanks for your reply. The question actually concerned expectation values of the harmonic oscillator using ladder operators and bras and kets
    How would I know that the probability distribution factorises and how does this lead to the first equation being correct ?
     
  5. May 26, 2017 #4

    vanhees71

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    It's always good to explain the example you have in mind in detail, so that we can answer questions precisely.

    If you have a 3D harmonic oscillator with the Hamiltonian
    $$\hat{H}=\frac{\hat{\vec{p}}^2}{2m} + \frac{m}{2} \sum_{j=1}^3 \omega_j^2 \hat{x}_j^2,$$
    obviously the number operators
    $$\hat{N}_j=\hat{a}_{j}^{\dagger} \hat{a}_j$$
    commute, and you can build an orthonormal basis of common eigenvectors (the Fock basis of three independent harmonic oscillators). Since each of the operators ##\hat{N}_j## only depend on ##\hat{x}_j## and ##\hat{p}_j## these states are obviously factorizing in the above described way, i.e., in the momentum representation you have
    $$u_{N_1,N_2,N_3}(\vec{p})=u_{N_1}(p_1) u_{N_2}(p_2) u_{N_3}(p_3).$$
    It's also clear that these are are complete set of energy eigenstates with eigenvalues
    $$E_{N_1,N_2,N_3}=\sum_{j=1}^3 \frac{\hbar \omega_j}{2} (2N_j+1).$$
     
  6. May 26, 2017 #5

    dyn

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    Thanks. The actual question is " calculate the following ground-state average for a 3-D harmonic oscillator < 0 | p12p22p32 | 0 > .
    The answer then states the word " factorisation" and states < p12p22p32 > = <p12> <p22 > <p32> and then each expectation value is calculated in the usual way. I would just like to know how to show this result using Dirac notation
     
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