(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The entire problem is quite in depth. But what I am having trouble with is just a small part of it, and it boils down to finding the following commutator:

[tex]\left[ S_{z}^{n},S_{y}\right][/tex]

where S_{z} and S_{y} are the quantum mechanical spin matrices.

The reason is that I have to commute S_{y} with an exponential that has S_{z} in it. So I expand the exponential as a series which contains S_{z}^{n}, so I need to find the above commutator.

2. Relevant equations

[tex]\left[ S_{z},S_{y}\right] =-i\hbar S_{x}[/tex]

[tex]\{ S_{x},S_{z}\}=0[/tex]

3. The attempt at a solution

[tex]S_{z}^{n}S_{y}=S_{z}^{n-1}(S_{z}S_{y})=S_{z}^{n-1}(S_{y}S_{z}-i\hbar S_{x})[/tex]

[tex]=S_{z}^{n-2}(S_{z}S_{y}S_{z}-i\hbar S_{z}S_{x})[/tex]

[tex]=S_{z}^{n-2}((S_{y}S_{z}-i\hbar S_{x})S_{z}-i\hbar S_{z}S_{x})[/tex]

[tex]=S_{z}^{n-2}(S_{y}S_{z}S_{z}-i\hbar (S_{x}S_{z}+S_{z}S_{x}))[/tex]

By the anticommutator relation for X and Z given above, the inner parenthesis is zero:

[tex]=S_{z}^{n-2}S_{y}S_{z}S_{z}[/tex]

This seems very strange though, otherwise it appears that if I keep doing this then as long as "n" is an even number, then [itex]\left[ S_{z}^{n},S_{y}\right][/itex] will commute.

and that the commutator will only be different from zero, with a value of [itex]-i\hbar S_{x}S_{z}^{n-1}[/itex] only if "n" is odd.

Is this right?

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

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!

# Spin commutators, [Sz^n, Sy]

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