Eigenspinor

In quantum mechanics, eigenspinors are thought of as basis vectors representing the general spin state of a particle.

Strictly speaking, they are not vectors at all, but in fact spinors.

For a single spin 1/2 particle, they can be defined as the eigenvectors of the Pauli matrices.

In the most general case, the eigenspinors for a system can be quite complicated.

If you have a collection of the Avogadro number of particles, each one with two (or more) possible spin states, writing down a complete set of eigenspinors would not be practically possible.

However, eigenspinors are very useful when dealing with the spins of a very small number of particles.

The simplest and most illuminating example of eigenspinors is for a single spin 1/2 particle.

A particle's spin has three components, corresponding to the three spatial dimensions:

Spin up is denoted as the column matrix:

Each component of the angular momentum thus has two eigenspinors.

By convention, the z direction is chosen as having the

The eigenspinors for the other two orthogonal directions follow from this convention:

: All of these results are but special cases of the eigenspinors for the direction specified by θ and φ in spherical coordinates - those eigenspinors are: Suppose there is a spin 1/2 particle in a state

To determine the probability of finding the particle in a spin up state, we simply multiply the state of the particle by the adjoint of the eigenspinor matrix representing spin up, and square the result.

Now, we simply square this value to obtain the probability of the particle being found in a spin up state:

Each set of eigenspinors forms a complete, orthonormal basis.

This means that any state can be written as a linear combination of the basis spinors.

The eigenspinors are eigenvectors of the Pauli matrices in the case of a single spin 1/2 particle.