Factorising expectation values

[dyn]

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

 

[vanhees71]

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
$$\langle p_1 \rangle^2=\left [\int_{\mathbb{R}} \mathrm{d} p_1 p_1 P(p_1) \right]^2 \neq \langle p_1^2 \rangle.$$

 

[dyn]

Thanks for your reply. The question actually concerned expectation values of the harmonic oscillator using ladder operators and bras and kets

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.

How would I know that the probability distribution factorises and how does this lead to the first equation being correct ?

 

[vanhees71]

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
$$E_{N_1,N_2,N_3}=\sum_{j=1}^3 \frac{\hbar \omega_j}{2} (2N_j+1).$$

 

[dyn]

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