- #1

Spinny

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Anyway, here's the problem. We're given the Dirac Hamiltonian [tex]H_D = \alpha_j p_j + \beta m[/tex], where [tex]p_j = -i\nabla_j[/tex] and the angular momentum components [tex]L_k = i\epsilon_{kln}x_lp_n[/tex]. We are then going to show the commutator relation

[tex] [H_D,L_k] = -i\epsilon_{kln}\alpha_lp_n[/tex]

Here's what I've got so far:

[tex][H_D,L,k]\psi = (H_DL_k-L_kH_D)\psi[/tex]

[tex]=(\alpha_j p_j+\beta m)i\epsilon_{kln}x_lp_n\psi -i\epsilon_{kln}x_lp_n(\alpha_jp_j+\beta m)\psi[/tex]

[tex]=\alpha_j p_j i\epsilon_{kln}x_lp_n\psi +\beta mi\epsilon_{kln}x_lp_n\psi -i\epsilon_{kln}x_lp_n\alpha_jp_j\psi-i\epsilon_{kln}x_lp_n\beta m\psi[/tex]

So far, (hopefully) so good. Now, as far as I can see, the second and the last part cancel, so we're left with

[tex][H_D,L_k]\psi = \alpha_j p_j i\epsilon_{kln}x_lp_n\psi - i\epsilon_{kln}x_lp_n\alpha_jp_j\psi[/tex]

Rewriting the first part we get

[tex]i\epsilon_{kln}p_n\alpha_j p_j(x_l\psi)[/tex]

and knowing that [tex]p_j[/tex] is a differential operator, we use the product rule, and get

[tex]i\epsilon_{kln}p_n(\psi \alpha_j p_j x_l + x_l\alpha_jp_j \psi)[/tex]

We then have that [tex]p_j x_l = -i[/tex] for j = l, and 0 for j != l. Thus

[tex]i\epsilon_{kln}p_n(-i\alpha_l\psi + x_l\alpha_jp_j\psi)= \epsilon_{kln}p_n\alpha_l \psi+i\epsilon_{kln}p_nx_l\alpha_jp_j\psi[/tex]

Putting this back in we get

[tex][H_D,L_k]\psi = \epsilon_{kln}p_n\alpha_l \psi+i\epsilon_{kln}p_nx_l\alpha_jp_j\psi- i\epsilon_{kln}x_lp_n\alpha_jp_j\psi[/tex]

But now the last two parts cancel, and we're left with

[tex][H_D,L_k]\psi = \epsilon_{kln}p_n\alpha_l \psi[/tex]

This is almost what I was supposed to get, only the factor -i is missing. It seems so close, so hopefully I'm not way off, but, where did I go wrong?!?