Collection of Solved Problems in Physics

Spin components and the square of spin magnitude

Task number: 4571

• Hint

  Recall the form of the spin‑\(\frac{1}{2}\) projection operators along the \(x, \, y, \, z\) axes expressed in terms of the Pauli matrices.

  Also recall the commutation relations between the components of the angular momentum operator.

• Solution to hint

  The spin‑\(\frac{1}{2}\) operator \(\hat{\vec{S}}\) has the form

  \[ \hat{\vec{S}} = \left ( \hat S_x, \hat S_y, \hat S_z \right ) = \frac{\hbar}{2} \left ( \hat \sigma_x, \hat \sigma_y, \hat \sigma_z \right ) = \frac{\hbar}{2} \left ( \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}, \begin{pmatrix} 0&−i \\ i&0 \end{pmatrix}, \begin{pmatrix} 1&0 \\ 0&−1 \end{pmatrix} \right ) \, , \]

  where \(\hat{\vec{\sigma}} = \left (\hat \sigma_x, \hat \sigma_y, \hat \sigma_z \right )\) are the Pauli matrices. Each component of \(\hat{\vec{S}}\) corresponds to the projection of the spin‑\(\frac{1}{2}\) along the respective axis.

  The commutation relations between the components of the angular momentum operator are \(\left [ \hat L_i, \hat L_j \right ] = i \hbar \varepsilon _{ijk} \hat L_k\). A detailed derivation of these commutation relations can be found in problem Commutators involving components of angular momentum, Solution j) – l).

• Solution a)

  To compare the forms of the commutation relations for the components of spin and those of angular momentum, we first compute the commutator of the spin components. Expressing them in terms of the Pauli matrices leads to

  \[ \left [ \hat S_i, \hat S_j \right ] = \left [ \frac{\hbar}{2} \hat \sigma_i, \frac{\hbar}{2} \hat \sigma_j \right ] = \frac{{\hbar}^2}{4} \left [ \hat \sigma_i, \hat \sigma_j \right ] \, . \]

  The Pauli matrices satisfy the commutation relations \(\left [ \hat{\sigma}_i, \hat{\sigma}_j \right ] = 2i \varepsilon_{ijk} \hat{\sigma}_k\). A detailed derivation is given in problem Pauliho matice, Řešení b) (Czech version only), and explicit products of pairs of Pauli matrices are listed in section Pauliho matice, Řešení a) (Czech version only). Substituting this relation and simplifying yields

  \[ \left [ \hat S_i, \hat S_j \right ] = \frac{{\hbar}^2}{4} 2i \varepsilon_{ijk} \hat \sigma_k = i \hbar \varepsilon_{ijk} \, \frac{\hbar}{2} \hat \sigma_k = i \hbar \varepsilon_{ijk} \hat S_k \, . \]

  Thus, the commutation relations for the spin‑\(\frac{1}{2}\) components take the form \(\left [ \hat S_i, \hat S_j \right ] = i \hbar \varepsilon_{ijk} \hat S_k\).

  Since the commutation relations for the components of angular momentum are \(\left [ \hat L_i, \hat L_j \right ] = i \hbar \varepsilon _{ijk} \hat L_k\), we have verified that the spin components fulfill the same commutation relations as the components of angular momentum.

• Solution b)

  To determine the spin angular momentum squared operator, we expand it as the sum of the squares of its components and rewrite these using the Pauli matrices

  \[ \hat S^2 = \hat S^2_x + \hat S^2_y + \hat S^2_z = \frac{\hbar ^2}{4} \left ( \hat \sigma^2_x + \hat \sigma^2_y + \hat \sigma^2_z \right ) \, . \]

  We now multiply out the squares of the Pauli matrices and add the results, obtaining

  \[ \hat S^2 = \frac{\hbar ^2}{4} \left [ \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix} \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix} + \begin{pmatrix} 0&−i \\ i&0 \end{pmatrix} \begin{pmatrix} 0&−i \\ i&0 \end{pmatrix} + \begin{pmatrix} 1&0 \\ 0&−1 \end{pmatrix} \begin{pmatrix} 1&0 \\ 0&−1 \end{pmatrix} \right ] = \] \[ = \frac{\hbar ^2}{4} \left [ \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} + \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} + \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} \right ] = \frac{3 \hbar ^2}{4} \begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} = \frac{3 \hbar ^2}{4} \hat{\mathbb E} \, . \]

  In summary, we obtain \(\hat S^2 = \frac{3 \hbar ^2}{4} \hat{\mathbb E}\), where \(\hat{\mathbb{E}}\) denotes the \(2 \times 2\) identity matrix.

• Answer

  a) The commutation relations for the spin components are \(\left [ \hat S_i, \hat S_j \right ] = i \hbar \varepsilon_{ijk} \hat S_k\). They therefore satisfy the same commutation relations as the components of angular momentum.

  b) The spin angular momentum squared operator is \(\hat S^2 = \frac{3 \hbar ^2}{4} \hat{\mathbb E}\), where \(\hat{\mathbb E}\) denotes the \(2 \times 2\) identity matrix.