Explicit description of spin-1/2

The Hilbert space of a spin-1/2 particle is two-dimensional. Moreover, $m_s$, the $S_z$ eigenvalue, can take on two values $-1/2$ and $1/2$. Therefore, $S_z$ should be represented by a 2$\times$2 matrix. If we choose to work in a basis in which $S_z$ is diagonal, then as a matrix, it will be given by

$$S_z={\left(\matrix{{\hbar \over 2} & 0 \cr 0 & -{\hbar \over 2}}\right)}$$

The two eigenvectors are

$$\left\vert m_s = {1 \over 2}\right> \equiv \left\vert{1 \over 2}\right>. = {\left(\matrix{1 \cr 0}\right)}$$

$$\left\vert m_s = -{1 \over 2}\right> \equiv \left\vert-{1 \over 2}\right> = {\left(\matrix{0 \cr 1}\right)}$$

These clearly satisfy

$${\left<{1 \over 2}\right.}{\left\vert{1 \over 2}\right>} = {\left<-{1 \over 2}\right.}{\left\vert-{1 \over 2}\right>}= 1$$

$${\left<{1 \over 2}\right.}{\left\vert-{1 \over 2}\right>} = {\left<-{1 \over 2}\right.}{\left\vert{1 \over 2}\right>}= 0$$

Thus, they form an orthonormal set of vectors and span the Hilbert space ${\cal H}_s$. The other spin operators are

$$S_x={\left(\matrix{0 & {\hbar \over 2} \cr {\hbar \over 2} &... ...matrix{0 & -{i\hbar \over 2} \cr {i\hbar \over 2} & 0}\right)}$$

It is straightforward to show that these satisfy the commutation relations ${\bf S}\times {\bf S}= i\hbar {\bf S}$ and both have eigenvalues $\pm \hbar/2$.

Let us use these to calculate the Casimir operator $S^2$:

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

$$= {\hbar^2 \over 4}\left[{\left(\matrix{0$$

$$= {\hbar^2 \over 4}\left[{\left(\matrix{1$$

$$= {\left(\matrix{{3 \over 4}\hbar^2$$

Note, however, that

$$s(s+1)\hbar^2 = {1 \over 2}\left({1 \over 2} + 1\right)\hbar^2 = {3 \over 4}\hbar^2$$

Also, ${\left\vert{1 \over 2}\right>}$ and ${\left\vert-{1 \over 2}\right>}$ are eigenvectors of $S^2$, both with eigenvalues $3\hbar^2/4$. Finally, since

$$S^2 = {3 \over 4}\hbar^2{\left(\matrix{1 & 0 \cr 0 & 1}\right)}$$

it is clear that $S^2$ commutes with $S_x$, $S_y$ and $S_z$ as expected for a Casimir operator.

Thus, the spin states are most generally denoted as

$$\vert s\;m_s\rangle = {\left\vert{1 \over 2}\;\;\;{1 \over 2}\right>}= {\left(\matrix{1 \cr 0}\right)}$$

$$= {\left\vert{1 \over 2}\;\;\;-{1 \over 2}\right>}= {\left(\matrix{0 \cr 1}\right)}$$

It is also possible to change from one spin state to another by means of the raising and lowering operators, which are defined by

$${\hbar\left(\matrix{0 1 \cr 0 0{\right){$$

$${\hbar\left(\matrix{0 0 \cr 1 0{\right){$$

which satisfy the following commutation rules:

$$\left[S_+,S_-\right] = 2S_z$$

$$\left[S_+,S_z\right] = -{\hbar \over 2}S_-$$

$$\left[S_-,S_z\right] = {\hbar \over 2}S_+$$

and additionally:

$$S^2 = {1 \over 2}\left(S_+S_- + S_- S_+\right) + S_z^2$$

$$= {1 \over 2}\left[S_+,S_-\right]_+ + S_z^2$$

where $\left[A,B\right]_+$ denotes the anticommutator between $A$ and $B$ defined to be $AB+BA$.

The action of the raising ($S_+$) and lowering ($S_-$) operators on the states can be worked out straightforwardly:

$$S_+{\left\vert{1 \over 2}\;\;\;{1 \over 2}\right>} = {\hbar\left(\matrix{0$$

$$S_+{\left\vert{1 \over 2}\;\;\;-{1 \over 2}\right>} = {\hbar\left(\matrix{0$$

$$S_-{\left\vert{1 \over 2}\;\;\;-{1 \over 2}\right>} = {\hbar\left(\matrix{0$$

$$S_-{\left\vert{1 \over 2}\;\;\;{1 \over 2}\right>} = {\hbar\left(\matrix{0$$

Generally, for spin-1/2,

$$S_+\vert s\;m_s\rangle = \hbar\vert s\;m_s+1\rangle$$

$$S_-\vert s\;m_s\rangle = \hbar\vert s\;m_s-1\rangle$$

It is customary to define the spin operators in terms of the so called Pauli matrices

$$\sigma_x = {\left(\matrix{0 & 1 \cr 1 & 0}\right)}\;\;\;\;\;... ...t)}\;\;\;\;\;\sigma_z={\left(\matrix{1 & 0 \cr 0 & -1}\right)}$$

such that

$${\bf S}= {\hbar \over 2}{\bf\sigma}$$

The Pauli matrices satisfy the following properties

1.

$${\left(\matrix{0 & 1 \cr 1 & 0}\right)}^2 = {\left(\matrix{0... ...ght)}^2 = {\left(\matrix{1 & 0 \cr 0 & -1}\right)}^2 = {\rm I}$$

2.

Anticommutation:

$$\left[\sigma_i,\sigma_j\right]_+=0\;\;\;\;\;\;\;\;\;\;{\rm for\;\;}i\neq j$$

3.

Commutation:

$${\bf\sigma}\times {\bf\sigma} = 2i{\bf\sigma}$$

4.

Cyclic multiplication:

$$\sigma_x\sigma_y = i\sigma_z\;\;\;\;\;\sigma_y\sigma_z = i\sigma_x\;\;\;\;\; \sigma_z\sigma_x = i\sigma_y$$

5.

Tracelessness:

$${\rm Tr}(\sigma_i) = 0$$

6.

Determinant:

$${\rm det}(\sigma_i) = -1$$

7.

The Pauli matrices plus the identity matrix form a kind of ``basis'' in the space of all possible 2$\times$2 matrices. Thus, any 2$\times$2 matrix, ${\rm A}$ can be expressed as a linear combination of ${\bf\sigma}$ and $I$:

$${\rm A} = \alpha_0 {\rm I} + \sum_{j=x,y,z}\alpha_j \sigma_j$$

where $\alpha_0$, $\alpha_x$, $\alpha_y$ and can, in general, be complex numbers.

8.

Finally, for two arbitrary vectors and , the following identity can be proven: