Rotating eigenstates of J operator into each other?

In summary, the conversation discusses the possibility of rotating eigenstates of a spin-J particle into each other. It is unclear whether this rotation refers to a unitary transformation or a rotation in 3D space. The participant suggests a possible unitary transformation using the identity matrix and the m states, but it is uncertain if this is considered a rotation. It is also noted that such rotations may result in superpositions of m states.
  • #1
QualTime
3
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Homework Statement



Consider the following set of eigenstates of a spin-J particle:

[tex]
|j,j > , ... , |j,m > , ... | j , -j >
[/tex]

where
[tex]
\hbar^2 j(j+1) , \hbar m
[/tex]
are the eigenvalues of J^2 and Jz, respectively. Is it always possible to rotate these states into each other? i.e. given |j,m> and |j,m'>, is it always possible to find a unitary rotation operator U^j such that
[tex]
|j,m' > = U^{(j)} |j, m > [/tex]

***

Not too sure how to approach this problem, although given that
[tex]
U^\dagger |j,m' > = U^\dagger U |j,m > [/tex]
and
[tex]
< j,m' | j,m > = \delta_{mm'}
[/tex]
I would think that
[tex]
< j,m' | U^\dagger | j,m' > = 0
[/tex]

which doesn't seem right hence the answer would be no.

Also the fact that the rotation matrix times a given eigenstate is in general a linear combination of 2j+1 independent states of the form |j,m'> makes me doubtful as well.

Any help would be appreciated (this isn't an actual homework question but taken from a practice exam so feel free to go into as much detail as possible as that would be really helpful).
 
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  • #2
What do you mean by rotation? I think it is always possible to construct such a unitary operator. One could just do it explicitly.

[itex]U = \mathbf{I} - \mid j, m \rangle \langle j, m \mid - \mid j, m' \rangle \langle j, m' \mid + \mid j, m' \rangle \langle j, m \mid - \mid j, m \rangle \langle j, m' \mid[/itex]

I think the above would work. But do you consider that a rotation? I suppose one might consider all special unitary operators to be rotations in a sense (rotations in the Hilbert space). However I don't know if you mean rotations as in rotations in 3D space, that is ## U = e^{-i\theta \hat{\mathbf{n}} \cdot \mathbf{J}} ##. If this is what you mean, then I'm not sure of the answer. I know such rotations will often result in superpositions of m states, which is not what you want.
 
Last edited:
  • #3
Rotation is a bit of a misnomer indeed. I think unitary transformation is what is meant here. I'll try your first suggestion (just need to check it's indeed unitary) but it looks good to me. Thanks.
 

FAQ: Rotating eigenstates of J operator into each other?

1. What are eigenstates and how do they relate to the J operator?

Eigenstates are quantum states that represent the possible outcomes of a measurement of a physical property, such as energy or spin. The J operator is a mathematical representation of angular momentum, and it has a set of eigenstates that correspond to different values of angular momentum.

2. Why is it important to rotate eigenstates of the J operator into each other?

Rotating eigenstates of the J operator allows us to understand how angular momentum changes under different transformations, such as rotations or reflections. This is crucial in quantum mechanics, as many physical systems involve angular momentum and understanding its behavior is essential for predicting and interpreting experimental results.

3. How do you rotate eigenstates of the J operator into each other?

The specific mathematical procedure for rotating eigenstates of the J operator into each other depends on the system and the type of transformation being applied. In general, it involves using a rotation matrix or operator to transform the original eigenstates into the desired ones.

4. Can the J operator be used to rotate eigenstates of other operators?

Yes, the J operator can be used to rotate eigenstates of other operators, as long as they are related to angular momentum in some way. For example, the J operator can be used to rotate eigenstates of the orbital angular momentum operator, which describes the motion of particles in a potential field.

5. What is the significance of the eigenvalues of the J operator?

The eigenvalues of the J operator represent the possible values of angular momentum that can be measured in a physical system. They are important because they provide a basis for understanding and predicting the behavior of particles with angular momentum, and they are used to classify and identify different types of particles based on their properties.

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