Shankar 14.4.1 using Ehrenfest's Theorem

[Jolb]

Homework Statement

Show that if
$$H = -\gamma\mathbf{L\cdot B}$$, and B is position independent,

$$\frac{\mathrm{d} \left \langle \mathbf{L} \right \rangle}{\mathrm{d} t}=\left \langle \boldsymbol\mu \times\mathbf{B} \right \rangle=\left \langle \boldsymbol\mu \right \rangle\times\mathbf{B}$$

Here H is the Hamiltonian, B is the magnetic field, L is the angular momentum operator, mu is the magnetic moment, and gamma is the gyromagnetic ratio.

I want to solve this using Ehrenfest's theorem.

Homework Equations

$$\boldsymbol\mu=\gamma\mathbf{L}$$

$$[L_i,L_j]=i\hbar\epsilon_{ijk}L_k$$
(the fancy way of writing the usual angular momentum commutation relations)

and Ehrenfest's theorem:
$$\frac{\mathrm{d} \left \langle Q \right \rangle}{\mathrm{d} t} = \frac{i}{\hbar}\left \langle [H,Q] \right \rangle+\left \langle \frac{\partial Q}{\partial t} \right \rangle$$

The Attempt at a Solution

First of all, L has no explicit time dependence so the last term in Ehrenfest's theorem cancels. I seem to be getting a factor of 2 in [H,L] that shouldn't be there:

The x component of [H,L] is:
$$[H,L]_x=\gamma\left ( L_xL_xB_x+L_xL_yB_y+L_xL_zB_z - L_xB_xL_x-L_yB_yL_x-L_zB_zL_x \right )$$

$$=\gamma i\hbar\left ( L_zB_y-L_yB_z +L_zB_y-L_yB_z \right )$$
Edit: ^^^^ This is my mistake.$$= 2i\hbar\left ( \mu_zB_y-\mu_yB_z \right )=\frac{2\hbar}{i}(\boldsymbol\mu\times\mathbf{B})_x$$
and likewise for the other components.

Putting it together I get
$$\frac{\mathrm{d} \left \langle \mathbf{L} \right \rangle}{\mathrm{d} t}=\frac{i}{\hbar}\left \langle [H,L]\right \rangle=\frac{i}{\hbar}\left \langle \frac{2\gamma\hbar}{i}\mathbf{L}\times\mathbf{B} \right \rangle=\left \langle 2\boldsymbol\mu\times\mathbf{B} \right \rangle$$

and I'm left with that pesky two.

Where did I go wrong?Edit: I found my error and pointed out where I went wrong above.

That line should be
$$[H,\mathbf{L}]_x=\gamma\left ((L_xL_y-L_yL_x)B_y+(L_xL_z-L_zL_x)B_z \right )=\gamma \left (i\hbar L_zB_y-i\hbar L_yB_z \right )=\frac{\hbar}{i}(\boldsymbol\mu \times\mathbf{B})_x$$
then everything works.

 

[member38644]

Your mistake is in the first step where you wrote [H,L]_x. The correct expression should be [H,L]_x=\gamma\left ( L_xL_xB_x+L_xL_yB_y+L_xL_zB_z - L_xB_xL_x-L_yB_yL_x-L_zB_zL_x \right )=\gamma i\hbar\left ( L_zB_y-L_yB_z +L_zB_y-L_yB_z \right )= 2\gamma i\hbar\left ( \mu_zB_y-\mu_yB_z \right )=2\gamma\hbar(\boldsymbol\mu\times\mathbf{B})_x
The extra factor of 2 comes from the fact that [L_i,L_j]=i\hbar\epsilon_{ijk}L_k is a sum over k, so when you expand the commutator [H,L]_x, you will have two terms with the same coefficient.