• Support PF! Buy your school textbooks, materials and every day products Here!

QM- A bit of manipulation of expectation values.

  • Thread starter trv
  • Start date
  • #1
trv
73
0

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>
 

Answers and Replies

  • #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
 
  • #3
trv
73
0
Yep that helps. Thanks Maverick.
 

Related Threads for: QM- A bit of manipulation of expectation values.

  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
1K
Replies
2
Views
4K
  • Last Post
Replies
1
Views
1K
Replies
4
Views
2K
Replies
2
Views
832
Replies
1
Views
2K
Top