(Created page with "<m>\sigma</m> being the annihilation operator of a two-level system (one of the Pauli matrices), <m>\sigma^\dagger</m> its conjugate, <m>\mu,\nu,\eta,\theta</m> boolean variables...") |
m |
||
| Line 11: | Line 11: | ||
</center> | </center> | ||
| − | I sometimes | + | 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 }} | ||
<m>\sigma</m> being the annihilation operator of a two-level system (one of the Pauli matrices), <m>\sigma^\dagger</m> its conjugate, <m>\mu,\nu,\eta,\theta</m> boolean variables (0 or 1), the following general commutator reads in normal order:
<m> [{\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). </m>
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).