# Using operators and finding expectation value

1. Aug 28, 2013

### A9876

1. The problem statement, all variables and given/known data

The expectation value of the time derivative of an arbitrary quantum operator $\hat{O}$ is given by the expression:

d$\langle$$\hat{O}$$\rangle$/dt$\equiv$$\langle$d$\hat{O}$/dt$\rangle$=$\langle$∂$\hat{O}$/∂t$\rangle$+i/hbar$\langle$$[$$\hat{H}$,$\hat{O}$$]$$\rangle$​

Obtain an expression for $\langle$d$\hat{L}$x/dt+d$\hat{L}$y/dt$\rangle$ where $\hat{H}$=$\hat{H}$00Bz$\hat{L}$z/hbar

2. Relevant equations

$[$$\hat{L}$x,$\hat{L}$y$]$=i*hbar$\hat{L}$z
$[$$\hat{L}$y,$\hat{L}$z$]$=i*hbar$\hat{L}$x
$[$$\hat{L}$z,$\hat{L}$x$]$=i*hbar$\hat{L}$y

$[$A,B$]$=AB-BA

3. The attempt at a solution

$\langle$d$\hat{L}$x/dt+d$\hat{L}$y/dt$\rangle$=d$\langle$$\hat{L}$x+$\hat{L}$y$\rangle$
=$\frac{1}{ih}$ d$\langle$$[$$\hat{L}$y,$\hat{L}$z$]$+$[$$\hat{L}$z,$\hat{L}$x$]$$\rangle$/dt
=$\frac{1}{ih}$$\langle$ ∂$[$$\hat{L}$y,$\hat{L}$z$]$+$[$$\hat{L}$z,$\hat{L}$x$]$/∂t$\rangle$+$\frac{i}{hbar}$$\langle$$[$$\hat{H}$,$[$$\hat{L}$y,$\hat{L}$z$]$+$[$$\hat{L}$z,$\hat{L}$x$]$$]$$\rangle$

I'm not sure how to continue on from this

2. Aug 29, 2013

### clamtrox

Just calculate the commutators $[L_x, H]$ and $[L_y, H]$? You know that H0 should commute with all L's (why?), and the commutator with the extra term is easy to calculate.

3. Aug 30, 2013

### A9876

I calculated the commutators $[L_x, H]$ and $[L_y, H]$ but I don't see how this helps me answer the question. I also can't figured out why H0 should commute with all L's.

Also $\hat{H}$0=-hbar2/2mr2 $\{$$\frac{∂}{∂r}$(r2$\frac{∂}{∂r}$)+$\frac{1}{sinθ}$$\frac{∂}{∂θ}$(sinθ$\frac{∂}{∂θ}$)+$\frac{1}{sin squared θ}$$\frac{∂ squared}{∂\phi squared}$$\}$+V(r)

where the angular-dependent part of the Hamiltonian corresponds to the total angular momentum operator $\hat{L}$2

Last edited: Aug 30, 2013
4. Aug 31, 2013

### vela

Staff Emeritus
You were given that
$$\bigg\langle \frac{d\hat{O}}{dt} \bigg\rangle = \bigg\langle \frac{\partial \hat{O}}{\partial t} \bigg\rangle + \frac{i}{\hbar} \langle [\hat{H},\hat{O}] \rangle.$$ What do you get if you let $\hat{O} = \hat{L}_x$?

You should be able to prove that $\hat{L}_i$ commutes with $\hat{L}^2$ using the property [AB,C]=A[B,C]+[A,C]B and the commutation relations you listed above.

5. Aug 31, 2013

### A9876

But shouldn't I substitute $\hat{O}$=$\hat{L}$x+$\hat{L}$y instead?

Oh I understand now. Thanks

6. Aug 31, 2013

### vela

Staff Emeritus
You asked why you want to calculate $[\hat{H},\hat{L}_i]$. Do you see why?

7. Aug 31, 2013

### A9876

Yh I do. My final answer is

$\langle$d$\hat{L}$x/dt+d$\hat{L}$y/dt$\rangle$=$\langle$∂($\hat{L}$x+$\hat{L}$y)/∂t$\rangle$+$\frac{i}{hbar}$$\langle$iμ0Bz($\hat{L}$y - $\hat{L}$x)$\rangle$

Could I further simplify this?