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Shankar 14.4.1 using Ehrenfest's Theorem

  1. Mar 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that if
    [tex]H = -\gamma\mathbf{L\cdot B}[/tex], and B is position independent,

    [tex]\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}[/tex]

    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.

    2. Relevant equations
    [tex]\boldsymbol\mu=\gamma\mathbf{L}[/tex]

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

    and Ehrenfest's theorem:
    [tex]\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[/tex]


    3. 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:
    [tex][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 )[/tex]

    [tex]=\gamma i\hbar\left ( L_zB_y-L_yB_z +L_zB_y-L_yB_z \right )[/tex]
    Edit: ^^^^ This is my mistake.


    [tex]= 2i\hbar\left ( \mu_zB_y-\mu_yB_z \right )=\frac{2\hbar}{i}(\boldsymbol\mu\times\mathbf{B})_x[/tex]
    and likewise for the other components.

    Putting it together I get
    [tex]\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[/tex]

    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
    [tex][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[/tex]
    then everything works.
     
    Last edited: Mar 4, 2012
  2. jcsd
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