Spin ${1\over 2}$

Earlier, we showed that both integer and half integer angular momentum could satisfy the commutation relations but that there is no functional representation for the half integer type. Some particles have half integer internal angular momentum, also called spin. We will now develop a spinor representation for spin ${1\over 2}$. There are no coordinates $\theta$ and $\phi$ associated with internal angular momentum so the only thing we have is our spinor representation.

The usual basis states are the eigenstates of $S_z$. We know the eigenvalues are $+{1\over 2}\hbar$ and $-{1\over 2}\hbar$. It is easy to derive the matrix operators for spin.

$$\bgroup\color{black}S_x ={\hbar\over 2} \left(\matrix{0 &1\cr... ...z ={\hbar\over 2} \left(\matrix{1 &0\cr 0 &-1\cr}\right)\egroup$$

These satisfy the usual commutation relations from which we derived the properties of angular momentum operators.

The Pauli Spin Matrices, $\sigma_i$, are simply defined and have the following properties.

$$\begin{eqnarray*} & S_i \equiv {\hbar\over 2} \sigma_i. \\ & \vec{S}={\hbar\ove... ... [\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k \\ & \sigma_i^2=1 \end{eqnarray*}$$

They also anti-commute.

$$\begin{eqnarray*} \sigma_x\sigma_y=-\sigma_y\sigma_x\ \ \ \ \ \ \ & \sigma_x\s... ...ma_z\sigma_x \ \ \ \ \ \ \ & \sigma_z\sigma_y=-\sigma_y\sigma_z \end{eqnarray*}$$

The $\sigma$ matrices are the Hermitian, Traceless matrices of dimension 2. Any 2 by 2 matrix can be written as a linear combination of the $\sigma$ matrices and the identity.

Example: The expectation value of $S_x$.
Example: The eigenvectors of $S_x$.
Example: The eigenvectors of $S_y$.

The rotation operators, for rotations of the coordinate axes can be computed from the formula $R_i(\theta_i)=e^{iL_i\theta_i/\hbar}$.

$$\bgroup\color{black}R_z(\theta)=\left(\matrix{e^{i\theta/2}&0... ...2}\cr -\sin{\theta\over 2}&\cos{\theta\over 2}}\right)\egroup$$

Note that the operator for a rotation through $2\pi$ radians is minus the identity matrix for any of the axes (because ${\theta\over 2}$ appears everywhere). The surprising result is that the sign of the wave function of all fermions is changed if we rotate through 360 degrees.

As for orbital angular momentum ( $\vec{L}$), there is also a magnetic moment associated with internal angular momentum ( $\vec{S}$).

$$\bgroup\color{black}\vec{\mu}_{spin}=-{eg\over 2mc} \vec{S}\egroup$$

This formula has an additional factor of $g$, the gyromagnetic ratio, compared to the formula for orbital angular momenta. For pointlike particles, like the electron, $g$ has been computed in Quantum ElectroDynamics to be a bit over 2, $g=2+{\alpha\over\pi}+...$. For particles with structure, like the proton or neutron, $g$ is hard to compute, but has been measured. Because the factor of 2 from $g$ cancels the factor of 2 from $s={1\over 2}$, the magnetic moment due to the spin of an electron is almost exactly equal to the magnetic moment due to the orbital angular momentum in an $\ell=1$ state. Both are 1 Bohr Magneton, $\mu_B={e\hbar\over 2mc}$.

$$\bgroup\color{black}H=-\vec{\mu}\cdot\vec{B}={eg\hbar\over 4mc}\vec{\sigma}\cdot\vec{B}=\mu_B\vec{\sigma}\cdot\vec{B}\egroup$$

If we choose the $z$ axis to be in the direction of $B$, then this reduces to

$$\bgroup\color{black}H=\mu_BB\sigma_z.\egroup$$

Example: The time development of an arbitrary electron state in a magnetic field.
Example: Nuclear Magnetic Resonance (NMR and MRI).