Commutation relation to find Sx, Sy

[M. next]

We know how to find S_{x} and S_{y} if we used S_{+} and S_{-}, and after finding S_{x} and S_{y}, we can prove that

[S_{x}, S_{y}]= i\hbarS_{z} (Equation 1)
and
[S_{y}, S_{z}]= i\hbarS_{x} (Equation 2)
and
[S_{z}, S_{x}]= i\hbarS_{y} (Equation 3)

but can we, starting from Equations 1, 2, and 3 find Sx and Sy? Can we work in the opposite direction?

 

[dauto]

No Sx, Sy, and Sz are not uniquely defined by their commutation relations. At some point a choice of representation must be made. To make that point obvious, just replace Sx with Sy, Sy with Sz, and Sz with Sx. Now they are all different than before since they've been replaced by each other, but they still satisfy the original commutation relations. The standard Pauli matrices is just one among many equally valid possible representations.

 

[WannabeNewton]

Can we work in the opposite direction?

If you choose a specific representation then yes, finding the explicit form of the spin operators in the chosen representation amounts to algebraically solving the Lie algebra.

 

[M. next]

Thank you for the answers. WannabeNewton, could you explain what you meant with more details? How should I be working if that's the case?

 

[ChrisVer]

I think WannabeNewton said that by choosing a specific representation for S_{x,y,z} you can just calculate their commutator relations (~algebra) and reproduce equations 1,2,3.

 

[M. next]

Yes, but my question was if I could find either Sx, Sy, or Sz by commutation relation. More clearly, I meant can we starting from equations 1, 2, 3 and knowing Sz find Sx? Sy?

 

[WannabeNewton]

Again the actual method of determining the representations of $S_i$ depends on the representation itself. But we can do even better than what you asked for directly above.

Consider more generally the angular momentum generators $J_i$ which includes both spin and orbital angular momentum. We can use $[J_i,J_j] = i\epsilon_{ijk}J_k$ (and only this) to find all angular momentum representations of the $J_i$ i.e. we get the complete set of simultaneous eigenstates $|j,m\rangle$ of $J^2$ and $J_z$ using only the above commutator relations and the method of raising and lower operators and then for each fixed $j = 0,\frac{1}{2},1,...$ we get an angular momentum representation of $J_i$ by calculating $\langle j',m' |J_i |j,m \rangle$ where $m = -j,...,j$. For the $j = \frac{1}{2}$ representation of the $S_i$ this gives us back the Pauli (spin) matrices. We can thus find all possible angular momentum matrix representations of the $J_i$ using only $[J_i,J_j] = i\epsilon_{ijk}J_k$ through the solving of a simultaneous eigenvalue problem for the compatible observables.

 

[ChrisVer]

Knowing Sz in one representation, you can find Sx,Sy in that too...either from eqs 1,2,3 or by finding the ladder operators S+ S-