Spin matrix representation in any arbitrary direction

[PhysicsTruth]

Homework Statement

Find the representation in which the component of spin along a direction $\hat{n}$ is diagonal, where $\hat{n}$ is a vector in the $x-z$ plane, making an angle $\theta$ with the $z$ axis.

Relevant Equations

$\sigma \cdot \hat{n} = \begin{pmatrix} cos(\theta) & sin(\theta)e^{-i\phi} \\ sin(\theta)e^{i\phi} & -cos(\theta) \end{pmatrix}$

$\sigma _z = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$

I've tried to use the 1st equation as a matrix to determine, but it clearly isn't a diagonal matrix. My guess is that I need to find the spin matrix along the direction $\hat{n}$, but do I need to find the eigenstates of $\sigma \cdot \hat{n}$ first and check if they form a diagonal matrix or not? Can someone help me in figuring out how to proceed?

 

[PeroK]

You could find the eigenvectors of that matrix, but other than that I'm not sure what the question is asking.

 

[PeroK]

The matrix you've found would then be diagonal in the basis of its eigenvectors. That might be what is intended.

 

[PhysicsTruth]

The matrix you've found would then be diagonal in the basis of its eigenvectors. That might be what is intended.

To be honest, the matrix doesn't have non-trivial solutions explicitly. It's only when one of the values is decided can the other value be determined. So, it might be diagonal only for a particular value of $\theta$ or $\phi$. Also, $\phi$ hasn't been mentioned explicitly, so can't really proceed with that even.

 

[PeroK]

To be honest, the matrix doesn't have non-trivial solutions explicitly. It's only when one of the values is decided can the other value be determined. So, it might be diagonal only for a particular value of $\theta$ or $\phi$. Also, $\phi$ hasn't been mentioned explicitly, so can't really proceed with that even.

It's manifestly Hermitian so must have an orthonormal eigenbasis. Moreover, it's physically identical to $S_z$ etc. So must have an eigenbasis.

 

[vanhees71]

Another way is to think about how rotations act on spinors an tgen use the rotation which maps $\vec{e}_z$ to $\vec{n}$.

 

[PhysicsTruth]

Another way is to think about how rotations act on spinors an tgen use the rotation which maps $\vec{e}_z$ to $\vec{n}$.

Okay, so the rotation around any unit axis $\hat{n}$ of a spinor is given by-
$Icos(\theta /2) + i(\hat{n} \cdot \sigma)sin(\theta /2)$
So here, it's getting rotated about the $y$ axis, so should I get the representation as required if I use this rotation form?

 

[PhysicsTruth]

Can someone just tell me how to write a Spin Matrix along any vector rotated by an angle $\theta$ from $z$ axis in terms of the eigenbasis of $\sigma _z$ Pauli spin matrix? Is it $cos(\theta /2) |+> + sin(\theta /2) |->$, where $|+>, |->$ are eigen vectors of $\sigma _z$ ?

 

[PhysicsTruth]

Can someone just tell me how to write a Spin Matrix along any vector rotated by an angle $\theta$ from $z$ axis in terms of the eigenbasis of $\sigma _z$ Pauli spin matrix? Is it $cos(\theta /2) |+> + sin(\theta /2) |->$, where $|+>, |->$ are eigen vectors of $\sigma _z$ ?

In that case, I can just perform an Unitary transformation to make it diagonal. So I need to know this.

 

[PeroK]

Can someone just tell me how to write a Spin Matrix along any vector rotated by an angle $\theta$ from $z$ axis in terms of the eigenbasis of $\sigma _z$ Pauli spin matrix? Is it $cos(\theta /2) |+> + sin(\theta /2) |->$, where $|+>, |->$ are eigen vectors of $\sigma _z$ ?

You already have the matrix in your original post. For the specific case of a direction in the x-z plane you have $\phi =0$.

 

[PeroK]

Can someone just tell me how to write a Spin Matrix along any vector rotated by an angle $\theta$ from $z$ axis in terms of the eigenbasis of $\sigma _z$ Pauli spin matrix? Is it $cos(\theta /2) |+> + sin(\theta /2) |->$, where $|+>, |->$ are eigen vectors of $\sigma _z$ ?

Yes, that's the eigenvector corresponding to eigenvalue $1$. You need the second eigenvector corresponding to eigenvalue $-1$.

 

[PeroK]

In that case, I can just perform an Unitary transformation to make it diagonal. So I need to know this.

You don't need to calculate the transformation to the eigenbasis, as you know it takes the matrix takes the same form as the Pauli matrix for $S_z$ in that basis.

 

[PhysicsTruth]

So, would the representation just be $\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$ in that case?

 

[PhysicsTruth]

So, would the representation just be $\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$ in that case?

I did get this after a wholesome unitary transformation to the new eigenbasis, which is the same as the $\sigma _z$ matrix.

 

[PeroK]

So, would the representation just be $\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$ in that case?

Yes, once you have the eigenvalues and vectors you know the matrix in that eigenbasis. Although, it can't hurt to grind out the transformation just to check!

 

[PhysicsTruth]

Yes, once you have the eigenvalues and vectors you know the matrix in that eigenbasis. Although, it can't hurt to grind out the transformation just to check!

Thanks. I have some problem to understand this intuitively. Can you just explain a bit as to why we end up getting this sole matrix when we change our basis and fix it along a direction?

 

[PeroK]

Thanks. I have some problem to understand this intuitively. Can you just explain a bit as to why we end up getting this sole matrix when we change our basis and fix it along a direction?

The physical explanation is that the z-axis is an arbitrary choice. Any other direction is physically equivalent (and, indeed, could be chosen as the z-axis). The formalism of spin about any direction must be equivalent to the formalism about the z-axis.

Mathematically, the action of an operator is completely determined by its action on a basis. And, the action on its eigenbasis is "diagonal". You don't have to transform bases, you can study the action of the operator on that eigenbasis directly.