[NOTES/QM-20001] Spin as a Dynamical Variable

Contents

1. What is spin?
2. What led to discovery of spin?
3. How is spin described in quantum mechanics?
4. Properties of spin operators.

What is spin?

• Spin of a particle is its angular momentum at rest.
• For a composite system, such as a nucleus, it is the value of angular momentum in the center of mass frame.
• For a classical point particle the angular momentum, given by $\vec{r}\times\vec{p} $, is zero when the particle is at rest. Therefore spin has no classical analogue.

What led to discovery of spin?

In order to explain anomalous Zeeman effect it was suggested by Goudsmidt and Uhlenbeck that electron possesses angular momentum at rest whose component in any fixed direction can take one of the two values ${1\over2}\hbar$ or $-{1\over 2}\hbar$). Associated with spin there is a magnetic moment, of one negative Bohr magneton, given by $$ \vec{\mu} = -{e\over mc}\vec{S} $$ Many elementary particles are found to have angular momentum at rest. This angular momentum is called spin.

How is spin described in quantum mechanics?

Spin is an observable associated with all the fundamental particles, having the same properties as the angular momentum. Therefore, we associate three operators $S_x,S_y,$ and $S_z$ with spin angular momentum and assume that they satisfy angular momentum algebra. \begin{eqnarray} [S_x,S_y]&=&i\hbar S_z \label{E1}\\{} [S_y,S_z]&=&i\hbar S_x \label{E2}\\{} [S_z,S_x]&=&i\hbar S_y \label{E3} \end{eqnarray}

Properties of spin operators.

\begin{frame} \frametitle{Properties of spin ... ... ... 1/3} The commutation relations of spin operators imply that the operator $\vec{S}^2=S_x^2+S_y^2+S_z^2$ commutes with all the three components of spin. Since different components of spin do not commute, a commuting set of operators has $\vec{S}^2$ and components of the spin along any one direction; most common choice being $\vec{S}^2$ and $S_z$.

The results on angular momentum apply to the spin also and we have

• The eigenvalues of $\vec{S}^2$ are given by $s(s+1)\hbar^2$ where $s$ is a positive integer of half integer.
• For a given value of $s$, the eigenvalues of $S_z$ are $s,s-1,s-2,\ldots,-s$.
• A particle will be said to have spin $s$ if the the maximum allowed value of $S_z$ is $s\hbar$, which is same as $\vec{S}^2$ having value $s(s+1)\hbar^2$.

A simultaneous eigenvector of $\vec{S}^2$ and $S_z$ will be denoted by $\ket{s m}$ which will have the properties \begin{eqnarray} \vec{S}^2 \ket{sm} &=& s(s+1)\hbar^2\ket{sm} \label{E4}\\ \vec{S}_z\ket{sm} &=& m \hbar \ket{sm} \label{E5} \end{eqnarray} In all there are $(2s+1)$ values of $m$ ranging from $-s$ to $s$ and therefore $(2s+1)$ eigenvectors $\ket{sm}$. The vector space needed to describe spin is linear span of all the vectors $\ket{sm}$ and is $(2s+1)$ dimensional