Eigenspinor

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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.

1 General eigenspinors
2 The spin 1/2 particle
3 Example usage
4 Properties
5 See also
6 References

[edit] General eigenspinors

In quantum mechanics, the spin of a particle or collection of particles is quantized. In particular, all particles have either half integer or integer spin. In the most general case, the eigenspinors for a system can be quite complicated. If you have a collection of Avogadro's number of particles, each one with two (or more) possible spin states, there would be no hope of writing down a complete set of eigenspinors. However, eigenspinors are very useful when dealing with the spins of a very small number of particles.

[edit] The spin 1/2 particle

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: $S x$ , $S y$ , and $S z$ . For a spin 1/2 particle, there are only two possible eigenstates of spin: spin up, and spin down. Spin up is denoted as the column matrix: $\chi_+ = \begin{bmatrix} 1\\ 0\\ \end{bmatrix}$ and spin down is $\chi_- = \begin{bmatrix} 0\\ 1\\ \end{bmatrix}$ .

Each component of the angular momentum thus has two eigenspinors. By convention, the z direction is chosen as having the $χ +$ and $χ -$ states as it's eigenspinors. The eigenspinors for the other two directions follow from this convention:

$S z$ :

$\chi_+^z = \begin{bmatrix} 1\\ 0\\ \end{bmatrix}$

$\chi_-^z = \begin{bmatrix} 0\\ 1\\ \end{bmatrix}$

$S x$ :

$\chi_+^x = {1 \over \sqrt{2}} \begin{bmatrix} 1\\ 1\\ \end{bmatrix}$

$\chi_-^x = {1 \over \sqrt{2}} \begin{bmatrix} 1\\ -1\\ \end{bmatrix}$

$S y$ :

$\chi_+^y = {1 \over \sqrt{2}} \begin{bmatrix} 1\\ i\\ \end{bmatrix}$

$\chi_-^y = {1 \over \sqrt{2}} \begin{bmatrix} 1\\ -i\\ \end{bmatrix}$

[edit] Example usage

Suppose there is a spin 1/2 particle in a state $\chi = {1 \over \sqrt{5}} \begin{bmatrix} 1\\ 2\\ \end{bmatrix}$ . To determine the probability of finding the particle in a spin up state, we simply multiply the state of the particle by the eigenspinor representing spin up, and square the result. Thus, the eigenspinor allows us to sample the part of the particle's state that is in the same direction as the eigenspinor. First we multiply the two eigenspinors: