Amending the postulates of quantum mechanics to include spin

It was Pauli who formalized the incorporation of spin into the non-relativistic framework of quantum mechanics. In order to do this, he needed to supplement the fundamental postulates of quantum mechanics with a few new ones. These are discussed below:

1.

To the position ${\bf R}$ and momentum ${\bf P}$ operators describing a particle, we now add the variable ${\bf S}$, which is a vector of operators

$${\bf S}= (S_x,S_y,S_z)$$

These operators satisfy the commutation rules of an angular momentum, namely

$$\left[S_x,S_y\right] = i\hbar S_z$$

$$\left[S_y,S_z\right] = i\hbar S_x$$

$$\left[S_z,S_x\right] = i\hbar S_y$$

This is an example of a Lie algebra. More generally, a Lie algebra is defined by the condition that a set of $n$ operators $X_1,...,X_n$ obey commutation relations of the form

$$[X_i,X_j]= \sum_i C_{ijk}X_k$$

In the above, the operators $X_i$ constitute the generators of a Lie group and the constants $C_{ijk}$ are called the structure constants of the group.

In the case of the spin operators, the commutation relations can be written compactly in terms of a vector cross product as

$${\bf S}\times {\bf S}= i\hbar {\bf S}$$

which can also be written as

$$[S_i,S_j]= i\hbar \sum_k \epsilon_k S_k$$

Here, $\epsilon_{ijk}$ is called the Levi-Civita tensor and is defined by

$$\epsilon_{ijk} \;\;\; = \;\;\;1\;\;\;\;\;\;\;\;\;\;{\rm if\;\;}ijk\;\; {\rm is\ a\ cyclic\ permutation\ of\ 123}$$

$$\;\;\; = \;\;\;-1\;\;\;\;\;\;\;\;{\rm if\;\;}ijk\;\; {\rm is\ a\ cyclic\ permutation\ of\ 321}$$

$$\;\;\; = \;\;\;0\;\;\;\;\;\;\;\;\;{\rm otherwise}$$

The $\epsilon_{ijk}$ are the structure constants of the spin Lie algebra.

2.

Like an angular momentum, there will be a Hilbert space ${\cal H}_s$ associated with spin which supports the spin eigenstates. Like an angular momentum, a set of basis vectors for this Hilbert space can be constructed out of simultaneous eigenvectors of the operators $S^2 = \vert{\bf S}\vert^2$ and $S_z$. The associated eigenvalues $s$ and $m_s$ satisfy the eigenvalue equations

$$S^2 \vert s\;m_s\rangle = s(s+1)\hbar^2 \vert s\;m_s\rangle$$

$$S_z \vert s\;m_s\rangle = m_s\hbar \vert s\;m_s\rangle \nonumber$$

Here, $\vert s\;m_s\rangle$ are the simultaneous eigenstates which satisfy

$$\langle s\;m_s\vert s\;m_{s'}\rangle = \delta_{m_s m_s'}$$

Here, also,

$$S^2 = S_x^2 + S_y^2 + S_z^2$$

satisfies

$$\left[S^2,S_z\right] = \left[S^2,S_y\right] = \left[S^2,S_z\right]=0$$

and is an example of what is called a Casimir operator. Generally, a Casimir operator is any operator which commutes with all members of the group. In the case, the members are just $S_x$, $S_y$ and $S_z$.

3.

The spin Hilbert space ${\cal H}_s$ must be joined to the Hilbert space ${\cal H}_r$ associated with the classical variables ${\bf R}$ and ${\bf P}$. This is done by forming a direct product or tensor product space:

$${\cal H} = {\cal H}_r \bigotimes {\cal H}_s$$

so that the spin variables commute with the ${\bf R}$ and ${\bf P}$:

$$\left[S_i,R_j\right]=\left[S_i,P_j\right] = 0$$

Now it can be seen that complete sets of commuting observables (CSCOs) which are used to describe a particle, e.g.,

$$\left\{X,Y,Z,S^2,S_z\right\}\;\;\;\;\;\;\;\;\left\{P_x,P_y,P_z,S^2,S_z\right\}$$

or, for spherically symmetric potentials:

$$\left\{H,L^2,L_z,S^2,S_z\right\}$$

where $H$ is the Hamiltonian. Thus, now five observables are needed to describe a particle. This means that there will be five associated quantum numbers. For a spherically symmetric potential independent of spin, these would then be $n,l,m,s,m_s$.

4.

The spin of a particle can take on half-integer values. In particular, the electron is a spin-${1 \over 2}$ particle ($s=1/2$) with two $m_s$ values $-1/2$ and $1/2$ so that, for an electron, the Hilbert space is two-dimensional.