Eigenspinor

In

.

General eigenspinors

In quantum mechanics, 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 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: ${\displaystyle S_{x}}$, ${\displaystyle S_{y}}$, and ${\displaystyle S_{z}}$. For a spin 1/2 particle, there are only two possible

of spin: spin up, and spin down. Spin up is denoted as the column matrix: ${\displaystyle \chi _{+}={\begin{bmatrix}1\\0\\\end{bmatrix}}}$ and spin down is ${\displaystyle \chi _{-}={\begin{bmatrix}0\\1\\\end{bmatrix}}}$.

Each component of the angular momentum thus has two eigenspinors. By convention, the z direction is chosen as having the ${\displaystyle \chi _{+}}$ and ${\displaystyle \chi _{-}}$ states as its eigenspinors. The eigenspinors for the other two orthogonal directions follow from this convention:

${\displaystyle S_{z}}$:

${\displaystyle \chi _{+}^{z}={\begin{bmatrix}1\\0\\\end{bmatrix}}}$

${\displaystyle \chi _{-}^{z}={\begin{bmatrix}0\\1\\\end{bmatrix}}}$

${\displaystyle S_{x}}$:

${\displaystyle \chi _{+}^{x}={1 \over {\sqrt {2}}}{\begin{bmatrix}1\\1\\\end{bmatrix}}}$

${\displaystyle \chi _{-}^{x}={1 \over {\sqrt {2}}}{\begin{bmatrix}1\\-1\\\end{bmatrix}}}$

${\displaystyle S_{y}}$:

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

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

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:

${\displaystyle \chi _{+}={\begin{bmatrix}\cos(\theta /2)\\e^{i\varphi }\sin(\theta /2)\\\end{bmatrix}}}$

${\displaystyle \chi _{-}={\begin{bmatrix}\sin(\theta /2)\\-e^{i\varphi }\cos(\theta /2)\\\end{bmatrix}}}$

Example usage

Suppose there is a spin 1/2 particle in a state ${\displaystyle \chi ={1 \over {\sqrt {5}}}{\begin{bmatrix}1\\2\\\end{bmatrix}}}$. To determine the probability of finding the particle in a spin up state, we simply multiply the state of the particle by the adjoint of the eigenspinor matrix 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:

${\displaystyle c_{+}={\begin{bmatrix}1\ 0\\\end{bmatrix}}*\chi ={1 \over {\sqrt {5}}}}$.

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

${\displaystyle P_{+}={1 \over 5}}$

Properties

Each set of eigenspinors forms a

spinors.

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

See also

• Spin
• Spinor
• Eigenvector
• Pauli matrices

References

• Griffiths, David J. (2005) Introduction to Quantum Mechanics(2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall.
  ISBN 0-13-111892-7
  .
• de la Peña, Luis (2006). Introducción a la mecánica cuántica (3 edición). México DF: Fondo de Cultura Económica.
  ISBN 968-16-7856-7
  .