Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I am looking to find a closed-form formula for the following commutator

[itex][J_{-}^{n},J_{+}^{k}][/itex]

where the operators are raising and lowering operators of the [itex]\mathfrak{su}(2)[/itex] algebra for which [itex][J_{+},J_{-}]=2J_0[/itex] and [itex][J_{0},J_{\pm}]=\pm J_{\pm}[/itex]

I've already made some progress and I obtained the following relations, which can be proved by induction :

[itex][J_{-}^{n},J_{+}]=-2nJ_0J_{-}^{n-1}-n(n-1)J_-^{n-1}[/itex]

[itex][J_{-},J_{+}^{n}]=-2nJ_{+}^{n-1}J_0-n(n-1)J_+^{n-1}[/itex]

The two last relations can be used to find a recurrence relation for the object I wish to compute, but I am hoping for a closed form formula.

Any ideas ?

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Closed-Form for commutation relation between powers of raising and lowering operators

**Physics Forums | Science Articles, Homework Help, Discussion**