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| − | + | $\sigma$ being the annihilation operator of a two-level system (one of the Pauli matrices), $\sigma^\dagger$ its conjugate, $\mu,\nu,\eta,\theta$ boolean variables (0 or 1), the following general commutator reads in normal order: | |
| − | + | $$ | |
| − | + | ||
[{\sigma}^{\dagger\mu}\sigma^\nu,{\sigma}^{\dagger\eta}\sigma^\theta] | [{\sigma}^{\dagger\mu}\sigma^\nu,{\sigma}^{\dagger\eta}\sigma^\theta] | ||
= | = | ||
| Line 8: | Line 7: | ||
+\mu\eta(\nu-\theta){\sigma}^\dagger | +\mu\eta(\nu-\theta){\sigma}^\dagger | ||
+[(1-\mu)\nu\eta(1-\theta)-\mu(1-\nu)(1-\eta)\theta](1-2{\sigma}^\dagger\sigma). | +[(1-\mu)\nu\eta(1-\theta)-\mu(1-\nu)(1-\eta)\theta](1-2{\sigma}^\dagger\sigma). | ||
| − | + | $$ | |
| − | + | ||
I sometimes need this formula but always have to derive it again, which is very annoying ([[Media:Commutator-2pauli-matrices.nb|see this]]). (Unrelatedly, I also hope it is the last time I typeset something with texvc, as I plan to move to [[MathJax]]). | I sometimes need this formula but always have to derive it again, which is very annoying ([[Media:Commutator-2pauli-matrices.nb|see this]]). (Unrelatedly, I also hope it is the last time I typeset something with texvc, as I plan to move to [[MathJax]]). | ||
{{wl-publish: 2011-01-25 19:19:55 -0500 | Fabrice }} | {{wl-publish: 2011-01-25 19:19:55 -0500 | Fabrice }} | ||
$\sigma$ being the annihilation operator of a two-level system (one of the Pauli matrices), $\sigma^\dagger$ its conjugate, $\mu,\nu,\eta,\theta$ boolean variables (0 or 1), the following general commutator reads in normal order:
$$ [{\sigma}^{\dagger\mu}\sigma^\nu,{\sigma}^{\dagger\eta}\sigma^\theta] = \nu\theta(\eta-\mu)\sigma +\mu\eta(\nu-\theta){\sigma}^\dagger +[(1-\mu)\nu\eta(1-\theta)-\mu(1-\nu)(1-\eta)\theta](1-2{\sigma}^\dagger\sigma). $$
I sometimes need this formula but always have to derive it again, which is very annoying (see this). (Unrelatedly, I also hope it is the last time I typeset something with texvc, as I plan to move to MathJax).