(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?

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# Homework Help: Spin commutators, [Sz^n, Sy]

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