Tensor product: commutator for spin

[rsaad]

Homework Statement

[S² _total, S_z ∅1]

Homework Equations

S² _total = S²∅1+ 1∅S²+2(S_x∅S_x+S_y∅S_y+ S_z∅S_z)

The Attempt at a Solution

I calculated it in steps:
(1∅S_x ² +S_x ²∅1) * S_z ∅1
=[S²_x, S_z] ∅1 + S_z∅S_x ²
=-h_cut i (S_xS_y+S_yS_x)∅1 + S_z∅S_x ²

Is it correct way of doing it? I mean I just did one small segment of it. There is still a lot of calculation to be done. Is there any short method?

 

[mark726]

I would like to offer some clarification and suggestions for solving this problem. Firstly, the notation used in the initial post is not standard and may cause confusion. It is important to use standard notation when communicating scientific concepts.

Now, moving on to the problem itself, the first equation shown is the definition of the total spin operator, which is the sum of the spin operators for each particle in the system. The second equation is the commutation relation for the total spin operator and the z-component of the spin operator for particle 1. This is a useful relation to have when working with spin operators, but it is not directly related to the problem at hand.

To solve this problem, you will need to use the definition of the total spin operator and the commutation relations for the spin operators. The first step would be to expand the equation for S2 total using the definition given. Then, you can use the commutation relations to simplify the expression.

There is no shortcut for solving this problem, but there are some tips that may help. Firstly, make sure to keep track of the order of the spin operators, as they do not commute with each other. Also, remember that the commutator of two operators is equal to the negative of the commutator of the same two operators, but with the order reversed.

In summary, to solve this problem correctly, you will need to use the definition of the total spin operator and the commutation relations for spin operators. Make sure to use standard notation and keep track of the order of the operators. There is no shortcut for solving this problem, but being organized and following the steps outlined above should help make the calculation more manageable.