quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, eigenspinors are

basis vector In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as ...

s representing the general spin state of a particle. For a single spin 1/2 particle, they can be defined as the

eigenvectors In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...

of the

Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...

. As such, they are vectors mathematically but physics convention distinguishes vectors from

spinors In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...

by their transformation behavior.

General eigenspinors

In quantum mechanics, the

spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...

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 the

Avogadro number The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...

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:

S_x

S_y

, and

S_z

. For a spin 1/2 particle, there are only two possible

eigenstates In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...

of spin: spin up, and spin down. Spin up is denoted as the column matrix:

\chi_+ = \begin
 1\\
 0\\
                  \end

and spin down is

\chi_- = \begin
 0\\
 1\\
                  \end

. Each component of the

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

thus has two eigenspinors. By convention, the z direction is chosen as having the

\chi_+

and

\chi_-

states as its eigenspinors. The eigenspinors for the other two orthogonal directions follow from this convention:

S_z

: :

\chi_+^z = \begin
 1\\
 0\\
                  \end

\chi_-^z = \begin
 0\\
 1\\
                  \end

S_x

: :

\chi_+^x =  \begin
 1\\
 1\\
                  \end

\chi_-^x =  \begin
 1\\
 -1\\
                  \end

S_y

: :

\chi_+^y =  \begin
 1\\
 i\\
\end

\chi_-^y =  \begin
 1\\
 -i\\
\end

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: :

\chi_+ = \begin
 \cos (\theta/2)\\
 e^ \sin (\theta/2)\\
\end

\chi_- = \begin
\sin (\theta/2)\\
 -e^ \cos (\theta/2)\\
\end

Example usage

Suppose there is a spin 1/2 particle in a state

\chi =  \begin
 1\\
 2\\
                  \end

. 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:

c_+ = \begin
 1\ 0\\

                  \end
*\chi =

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

P_+ =

Properties

Each set of eigenspinors forms a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpe ...

basis. This means that any state can be written as a

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

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

Notes

References