Calculating eigenvalue and normalised eigenvector due to multiple rotated Stern-Gerlach Apparatusses

Question

Suppose a beam of particles of spin 1/2 is incident on a Stern-Gerlach apparatus aligned in the z-direction, which divides the beam into components with magnetic spin quantum numbers $m_s = 1/2$ . The $m_s = 1/2$ beam is incident on a Stern-Gerlach apparatus aligned in the direction

$e_\theta = \begin{bmatrix} sin(\theta) \\ 0 \\ cos(\theta) \end{bmatrix} $

Here I'm assuming that we work in the basis of $\hat S_z$. The spin operator $\hat S$ is represented by $(\hbar /2)\sigma$, where $\sigma$ is the usual vector of Pauli matrices

How do I find the eigenvalue and normalised eigenvectors of $e_\theta\cdot \hat S $ ?

The first road block I stumble upon, is that I'm unsure how to even address the problem of taking the product of a $3\times1$ and a $2\times2$ matrix. Also am I correct in thinking that $\hat S$ in this case is simply $\sigma_z$ ? Thanks in advance.

Answer 1

This is a common issue that appears when first learning about this, and what you have to do is separate the different types of vectors that are appearing. There are the "classical" vectors $e_{\theta}$ and $\vec{S}$ that have three components corresponding to three component directions in position space. Then there are the state vectors that live in an internal "spin" Hilbert space, and these are the vectors associated with Pauli matrices $\sigma_i$.

To be explicit, here's how I would write things down (leaving off factors of 2 and $\hbar$ and the like): $$\vec{S} = \hat{x}\sigma_x + \hat{y}\sigma_y + \hat{z}\sigma_z,$$ and $$\vec{e}_\theta = \hat{x}\sin\theta + \hat{y}(0)+ \hat{z}\cos\theta.$$ Then, the dot product $\vec{e}_\theta\cdot\vec{S}$ is interpreted to be the dot product of vectors in real space, and therefore $$\vec{e}_\theta\cdot\vec{S} = \sigma_x\sin\theta + \sigma_y(0) + \sigma_x\cos\theta.$$ This last quantity is clearly a 2 by 2 matrix representation of a spin operator. You can look up the matrix representation of the Pauli operators written in the eigenbasis of $S_z$, and so you can get a matrix that you can diagonalize directly.