Spin-½ ↩

shorthands: {}
aliases: [Spin-1/2, Spin-half, Half spin, spin-½]
created: 2022-01-06 08:52:14
modified: 2022-01-10 04:13:04

Overview

The spin operator for a half spin particle, like the electron:

The components of satisfy the relations of angular momentum operators. The components act on a 2D Hilbert space with the basis.

Where denotes the Pauli matrices.
The operator takes this form:

Mathematical formulation

Let the spin operator be . Now if is a unit vector, then according to the experimental evidences of the spin of the electron has only the two eigenvalues :

Without loss of generality, one can choose and introduce the following notation:

The eigenvalue equation then takes the form:

Since the spin is a physical observable, is a Hermitian operator, and the states and belonging to distinct eigenvalues are orthogonal (see this), so

We can normalize them to unity:

Since the spin is angular momentum, the components satisfy angular momentum commutation relations:

Where ¹ with the inversion:

For spin , has the eigenvalue :

Now we can act on these states with the ladder operators with and (see this):

We can now represent the spin operators in the basis of the states and by the spin matrices:

From this we can easily obtain the representation of the operators:

From this we can introduce the Pauli matrices:

We can see that the space of states is finite-dimensional here opposed to the usual case where the state is in an infinite dimensional Hilbert space.