SO(3) rotation of eigenvectors

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• spaghetti3451
In summary: And you have to use the spin-1/2 representation. So you should take the Pauli matrices. In general, you would take the spin-j representation and the corresponding matrices.In summary, the conversation discusses the change in eigenvectors of a quantum system described by a magnetic field ##\vec{B} = (0,0,B)## when the magnetic field is rotated to ##\vec{B} = (B\sin\theta\cos\phi,B\sin\theta\sin\phi,B\cos\theta)##. The eigenvectors are given by ##(\cos(\theta/2),e^{i\phi}\sin(\theta/2))## and ##(-\sin(\theta/
spaghetti3451
Consider the eigenvectors ##(0, 1)## and ##(1, 0)## for the quantum system described the magnetic field ##\vec{B} = (0,0,B)##.

Say I now rotate the magnetic field as ##\vec{B} = (B\sin\theta\cos\phi,B\sin\theta\sin\phi,B\cos\theta)##.

Then the eigenvectors are supposed to change as ##(\cos(\theta/2),e^{i\phi}\sin(\theta/2))## and ##(-\sin(\theta/2),e^{i\phi}\cos(\theta/2))##.

How can I prove that under SO(3) rotation, this is how the eigenvectors change?

P.S. : I've written the eigenvectors as row vectors, but they should be column vectors.

I was wondering if you could explain why a general rotation matrix is given by:

##\begin{pmatrix}
-\sin(\theta/2) & \cos(\theta/2) \\
e^{i\phi}\cos(\theta/2) & e^{i\phi}\sin(\theta/2)\\
\end{pmatrix}##?

Use the angles ##\theta## and ##\phi## to parametrize the 2x2 spin matrix ##S_u## (along axis u) and solve its eigenvectors. Both the 3D spatial rotation and the 2D state space rotation have two parameters in them.

Note that the full rotation group is a three-parameter group. Most common are two ways to parametrize it. One is to use a vector ##\vec{\phi}## in a sphere of radius ##\pi## (with diametral points on the boundary identified). Then the spin-1/2 representation is given by
$$\hat{D}(\vec{\phi})=\exp(-\mathrm{i} \vec{\phi} \cdot \hat{\vec{\sigma}}/2),$$
where ##\hat{\vec{\sigma}}## are the Pauli matrices.

The other way are Euler angles. In quantum theory one usually uses the variant, where you rotate around the 3- and the 2-axis:
$$\hat{R}(\alpha,\beta,\gamma)=\hat{D}_3(\alpha) \hat{D}_2(\beta) \hat{D}_3(\gamma), \quad \alpha,\gamma \in [0,2 \pi[, \quad \beta \in [0,\pi[.$$
The matrices ##\hat{D}_j## (##j \in \{1,2,3 \}##) are given by
$$\hat{D}_j(\phi)=\exp(-\mathrm{i} \sigma_j \phi/2).$$

You will find the matrices representing ##S_{x}, S_{y}## and ##S_z## from the site I linked to. Now create the spin operator ##S_u =
\sin\theta\cos\phi S_x + \sin\theta\sin\phi S_y + \cos\theta S_z## and calculate its eigenvalues and eigenvectors with the usual method of characteristic polynomial, linear pair of equations...

Last edited:
vanhees71 said:
Note that the full rotation group is a three-parameter group. Most common are two ways to parametrize it. One is to use a vector ##\vec{\phi}## in a sphere of radius ##\pi## (with diametral points on the boundary identified). Then the spin-1/2 representation is given by
$$\hat{D}(\vec{\phi})=\exp(-\mathrm{i} \vec{\phi} \cdot \hat{\vec{\sigma}}/2),$$
where ##\hat{\vec{\sigma}}## are the Pauli matrices.

In my case, I have a two-sphere. So, I was wondering which Pauli matrices correspond to ##\phi## and ##\theta##.

Also, what does it mean for the diametral points on the boundary to be identified?

I don't know, what you want to achieve with your rotation. If you want a point on the two-sphere the meaning of the usual angles (polar and azimuthal) is to rotate the point on the northpole first by ##\theta## (##\theta \in (0,\pi)##) around the ##x## axis and then the result by ##\phi## (##\phi \in [0,2 \pi)##) around the (new) ##z## axis, i.e.,
$$\hat{D}(\theta,\phi)=\hat{D}_3(\phi) \hat{D}_1(\theta).$$

1. What is the significance of SO(3) rotation of eigenvectors?

The SO(3) rotation of eigenvectors is a mathematical concept that is used to describe the rotation of a vector in three-dimensional space. It is an important concept in linear algebra and has many applications in physics and engineering.

2. How is SO(3) rotation of eigenvectors different from regular vector rotation?

SO(3) rotation of eigenvectors involves rotating a vector in three-dimensional space while maintaining its orientation, whereas regular vector rotation may change the direction or magnitude of the vector. SO(3) rotation is also a special type of orthogonal rotation, which preserves lengths and angles.

3. Can you explain the mathematical formula for SO(3) rotation of eigenvectors?

The SO(3) rotation of eigenvectors can be represented by a 3x3 rotation matrix, which is calculated using the eigenvectors and eigenvalues of the original vector. The formula for this rotation matrix involves trigonometric functions and is based on the principles of linear algebra.

4. How is SO(3) rotation of eigenvectors used in computer graphics?

In computer graphics, SO(3) rotation of eigenvectors is used to create smooth and realistic animations of 3D objects. By applying the rotation matrix to each vertex of a 3D model, the object can be rotated in a way that appears natural and visually appealing.

5. Are there any real-world applications of SO(3) rotation of eigenvectors?

SO(3) rotation of eigenvectors has many real-world applications, particularly in physics and engineering. It is commonly used in the study of rigid body dynamics, such as the rotation of planets and satellites. It is also used in robotics, computer vision, and navigation systems to calculate the orientation of objects in three-dimensional space.

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