QM- A bit of manipulation of expectation values.

[trv]

Homework Statement

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

Homework Equations

As above.

The Attempt at a Solution

(delta Qhat)²=<Qhat²-Qhat<Qhat>-<Qhat>Qhat+<Qhat>²>
L.H.S=<Qhat²-2Qhat<Qhat>+<Qhat>²>

 

[maverick_starstrider]

Right, you've got it

$\langle \hat{Q}^2 -2\hat{Q}\langle \hat{Q} \rangle +\langle \hat{Q} \rangle \rangle$

Now recognize that $\langle X \rangle$ 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. $\langle \langle \hat{Q} \rangle \rangle = \langle \hat{Q} \rangle$ and so on. Therefore you get:

$\langle \hat{Q}^2 \rangle - 2 \langle \hat{Q} \rangle \langle \hat{Q} \rangle + \langle \hat{Q} \rangle ^2$

and I assume you can take it from there

 

[trv]

Yep that helps. Thanks Maverick.