The problem asks to identify which operator does NOT commute with the dot product of orbital angular momentum ($ \vec{L}$) and spin angular momentum ($ \vec{S}$). Two operators commute if their commutator is zero. The commutator is defined as $ [A, B] = AB - BA $. We are interested in the operator $ \vec{L} \cdot \vec{S} = L_x S_x + L_y S_y + L_z S_z $.
Commutation Analysis
We need to check the commutation relation $ [\vec{L} \cdot \vec{S}, O] $ for each option $ O $. We use the standard commutation relations for angular momentum: $ [L_i, L_j] = i \hbar \epsilon_{ijk} L_k $, $ [S_i, S_j] = i \hbar \epsilon_{ijk} S_k $, and $ [L_i, S_j] = 0 $.
Option 1: $ S_z $
Let's calculate $ [\vec{L} \cdot \vec{S}, S_z] $:
$ [\vec{L} \cdot \vec{S}, S_z] = [L_x S_x + L_y S_y + L_z S_z, S_z] $Using linearity and the properties of commutators:
$ = [L_x S_x, S_z] + [L_y S_y, S_z] + [L_z S_z, S_z] $We know $ [L_z S_z, S_z] = 0 $ because $ [S_z, S_z] = 0 $.
For the other terms:
$ [L_x S_x, S_z] = L_x [S_x, S_z] + [L_x, S_z] S_x $Since $ [L_x, S_z] = 0 $ and $ [S_x, S_z] = -i \hbar S_y $:
$ = L_x (-i \hbar S_y) + (0) S_x = -i \hbar L_x S_y $Similarly:
$ [L_y S_y, S_z] = L_y [S_y, S_z] + [L_y, S_z] S_y $Since $ [L_y, S_z] = 0 $ and $ [S_y, S_z] = i \hbar S_x $:
$ = L_y (i \hbar S_x) + (0) S_y = i \hbar L_y S_x $Summing the terms:
$ [\vec{L} \cdot \vec{S}, S_z] = -i \hbar L_x S_y + i \hbar L_y S_x = i \hbar (L_y S_x - L_x S_y) $Since this result is non-zero, $ \vec{L} \cdot \vec{S} $ does NOT commute with $ S_z $.
Options 2, 3, and 4: $ L^2 $, $ S^2 $, and $ (\vec{L} + \vec{S})^2 $
Commutation with $ L^2 $: $ [\vec{L} \cdot \vec{S}, L^2] = [\sum_i L_i S_i, \sum_j L_j^2] $. Each term $ [L_i S_i, L_j^2] $ is zero because $ L_i $ commutes with $ L^2 $, and $ S_i $ acts on a different space and commutes with $ L^2 $. Thus, $ [\vec{L} \cdot \vec{S}, L^2] = 0 $.
Commutation with $ S^2 $: By symmetry, $ [\vec{L} \cdot \vec{S}, S^2] = 0 $. $ S_i $ commutes with $ S^2 $, and $ L_i $ commutes with $ S^2 $.
Commutation with $ (\vec{L} + \vec{S})^2 $: $ (\vec{L} + \vec{S})^2 = L^2 + S^2 + 2 \vec{L} \cdot \vec{S} $. The commutator is $ [\vec{L} \cdot \vec{S}, L^2 + S^2 + 2 \vec{L} \cdot \vec{S}] $. Since $ [\vec{L} \cdot \vec{S}, L^2] = 0 $, $ [\vec{L} \cdot \vec{S}, S^2] = 0 $, and $ [\vec{L} \cdot \vec{S}, 2 \vec{L} \cdot \vec{S}] = 0 $, the total commutator is $ 0 $.
Conclusion
The operator $ \vec{L} \cdot \vec{S} $ commutes with $ L^2 $, $ S^2 $, and $ (\vec{L} + \vec{S})^2 $. It does NOT commute with $ S_z $.
Important Questions from Angular Momentum Operators Eigenvalues Clebsch Gordan
- Consider two particles with angular momenta $j_1 = 2\hbar$ and $j_2 = \hbar/2$. If the expression
$$|j = 5/2, m = 3/2\rangle = \begin{cases} c_1|j_1 = 2, m_1 = 1\rangle|j_2 = 1/2, m_2 = 1/2\rangle + \\ c_2|j_1 = 2, m_1 = 2\rangle|j_2 = 1/2, m_2 = -1/2\rangle \end{cases}$$
gives an eigenstate of the total angular momentum of the two particles, using standard notation. Which of the following is true?
(Hint: $\hat{J}_{\pm}|j, m\rangle = \sqrt{j(j + 1) - m(m \pm 1)} |j, m \pm 1\rangle$) - A system of three non-identical spin $\frac{1}{2}$ particles has the Hamiltonian $H = \frac{A}{\hbar^2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3$, where $\vec{S}_1, \vec{S}_2$ and $\vec{S}_3$ are the spin operators of particles labelled $1,2$ and $3$ respectively and $A$ is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is
- $H$ is the Hamiltonian, $\vec{L}$ the orbital angular momentum and $L_z$ is the $z$-component of $\vec{L}$. The $1s$ state of the hydrogen atom in the non-relativistic formalism is an eigen function of which one of the following sets of operators?
- An atom with non-zero magnetic moment has an angular momentum of magnitude $\sqrt{12}\hbar$. When a beam of such atoms is passed through a Stern-Gerlach apparatus, how many beams does it split into?
- In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular momentum vector and the positive $z$ axis for a $2p$ electron?
