Rotation Operator: Spin 1/2 vs Spin 1

In summary, you can find the rotation operator for spin 1/2 by using the generators, and then exponentiating them.
  • #1
M. next
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How does finding the rotation operator for a spin 1/2 particle differ from finding that of a spin 1 particle?
 
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  • #2
if you have the generators (angular momentum operators) then its practically the same thing you just exponentiation.
Im not sure if that's what you're asking..
 
  • #3
Thanks for your reply, but I was pointing to a different road, my question in other words is how to derive, rotation operator for spin 1? How do we get there?
 
  • #4
M. next said:
Thanks for your reply, but I was pointing to a different road, my question in other words is how to derive, rotation operator for spin 1? How do we get there?

How did you find the rotation operator for spin 1/2? You probably started with the rules for the generators [itex]J_i[/itex], namely the commutation relation

[tex] [J_i,J_j] = i \epsilon_{ijk} J_k[/tex]

and the eigenvalue conditions

[tex] J^2 | j,m\rangle = j(j+1) | j,m\rangle,~~~J_3 | j,m\rangle = m | j,m\rangle .[/tex]

You then have to choose a set of basis vectors to correspond to the states [itex]| j,m\rangle[/itex]. You can choose a basis so that [itex]J_3[/itex] is diagonal. From here, you can use trial and error to find a pair of matrices that have the correct commutation relations with [itex]J_3[/itex] and satisfy the [itex]J^2[/itex] equation. Otherwise you can form the raising and lowering operators

[tex] J_\pm = \frac{1}{\sqrt{2}} (J_1\pm iJ_2),[/tex]

[tex] J_3 J_\pm | j,m\rangle = (m\pm 1) | j,m\pm 1\rangle[/tex]

and note that

[tex] J_+ | 1,1\rangle = J_- | 1,-1\rangle =0.[/tex]

These last conditions can be solved with less guesswork.

Once you have the generators, you can exponentiate them to find the rotation matrices.
 
  • #5
Can entangled spin 1/2 particles have time rate of change spins ? +-+-+-+- Or is spin a fixed value ?
 
  • #6
Thanks for your reply, but actually I didn't use this way.
We were trying to solve it without using the matrix-method. We know that rotation operator about some axis of unit vector u equals to exp[i/[itex]\hbar[/itex]*uSθ]
Then I can manipulate that and use Taylor's expansion to expand the exponential and then separate the terms into odd powers and even powers to end up with using again the Taylor's expansion but now to compile the 'odd' 'even' terms.

Then what procedure should be done to know spin rotation operator for spin equals 1?

If not what is the correct thing to do?

Thank you a lot.
 
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  • #7
It is simple,the generators here are like e(imθ) for rotation about z axis,where m=1/2 for spin 1/2 and m=1 for spin 1.i.e. replace θ/2 by θ.
 
  • #8
Thank you, i tried it and it worked! Thank you andrien.
 

FAQ: Rotation Operator: Spin 1/2 vs Spin 1

What is the difference between the rotation operator for spin 1/2 and the rotation operator for spin 1?

The rotation operator is a mathematical tool used to describe how an object or system changes when rotated in space. The main difference between the rotation operator for spin 1/2 and spin 1 is the number of degrees of freedom. Spin 1/2 particles have two possible spin states, while spin 1 particles have three possible spin states.

How does the rotation operator for spin 1/2 affect the spin state of a particle?

The rotation operator for spin 1/2 causes the spin state of a particle to change from one spin state to the other. This is known as spin flip. The spin flip occurs at a specific angle of rotation, which is determined by the properties of the particle.

Can the rotation operator for spin 1/2 be used to rotate spin 1 particles?

Yes, the rotation operator for spin 1/2 can be used to rotate spin 1 particles. However, this operator only describes the rotation of the spin component of the particle and does not account for other degrees of freedom.

How does the rotation operator for spin 1 affect the spin state of a particle?

The rotation operator for spin 1 causes the spin state of a particle to change from one spin state to another, just like the operator for spin 1/2. However, since spin 1 particles have three possible spin states, the rotation operator can cause the particle to flip to any of these states depending on the angle of rotation.

Is the rotation operator for spin 1/2 the same as the rotation operator for spin 1/2 in all reference frames?

No, the rotation operator for spin 1/2 is not the same in all reference frames. This is because the rotation operator depends on the orientation of the coordinate system. Therefore, the operator will change in different reference frames, even though the physical properties of the particle remain the same.

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