Commutation relation to find Sx, Sy

In summary: Sx,Sy.In summary, we can find Sx and Sy using Sz and the commutation relations [Sx, Sy] = iħSz, [Sy, Sz] = iħSx, and [Sz, Sx] = iħSy. However, Sx, Sy, and Sz are not uniquely defined by these relations and a specific representation must be chosen. This can be done by solving the Lie algebra and using the method of raising and lowering operators. With this method, we can find all possible representations of the angular momentum generators Jx, Jy, and Jz, including the familiar Pauli (spin) matrices.
  • #1
M. next
382
0
We know how to find S[itex]_{x}[/itex] and S[itex]_{y}[/itex] if we used S[itex]_{+}[/itex] and S[itex]_{-}[/itex], and after finding S[itex]_{x}[/itex] and S[itex]_{y}[/itex], we can prove that

[S[itex]_{x}[/itex], S[itex]_{y}[/itex]]= i[itex]\hbar[/itex]S[itex]_{z}[/itex] (Equation 1)
and
[S[itex]_{y}[/itex], S[itex]_{z}[/itex]]= i[itex]\hbar[/itex]S[itex]_{x}[/itex] (Equation 2)
and
[S[itex]_{z}[/itex], S[itex]_{x}[/itex]]= i[itex]\hbar[/itex]S[itex]_{y}[/itex] (Equation 3)

but can we, starting from Equations 1, 2, and 3 find Sx and Sy? Can we work in the opposite direction?
 
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  • #2
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 Now they are all different than before since they've been replaced by each other, but they still satisfy the original commutation relations.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.
 
  • #3
M. next said:
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.
 
  • #4
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?
 
  • #5
I think WannabeNewton said that by choosing a specific representation for [itex]S_{x,y,z}[/itex] you can just calculate their commutator relations (~algebra) and reproduce equations 1,2,3.
 
  • #6
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?
 
  • #7
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.
 
  • #8
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-
 

Related to Commutation relation to find Sx, Sy

1. What is the commutation relation for finding Sx and Sy?

The commutation relation for finding Sx and Sy is [Sx, Sy] = iℏSz, where Sx and Sy are the x and y components of the spin operator, and Sz is the z component.

2. How do you use the commutation relation to find Sx and Sy?

To use the commutation relation to find Sx and Sy, you first need to know the value of Sz. Then, simply plug in the values of Sx, Sy, and Sz into the relation [Sx, Sy] = iℏSz and solve for the unknown component.

3. Can the commutation relation be used to find other spin components?

Yes, the commutation relation [Sx, Sy] = iℏSz can also be used to find other spin components, such as Sz and Sx.

4. How does the commutation relation relate to the uncertainty principle?

The commutation relation is related to the uncertainty principle in that it provides a mathematical relationship between two non-commuting observables, which leads to the uncertainty principle. This means that the more precisely one spin component is known, the less precisely the other can be known.

5. Are there other commutation relations for finding spin components?

Yes, there are other commutation relations for finding spin components, such as [Sz, Sy] = iℏSx and [Sz, Sx] = iℏSy. These relations can be derived from the fundamental commutation relation [Sx, Sy] = iℏSz.

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