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
An electron in the Coulomb field of a proton is in the following state of coherent superposition of orthonormal states $\psi_{nlm}$
$\Psi = \frac{1}{3}\psi_{100} + \frac{1}{\sqrt{3}}\psi_{210} - \frac{\sqrt{5}}{3}\psi_{320}$
Let $E_1, E_2$, and $E_3$ represent the first three energy levels of the system. A sequence of measurements is done on the same system at different times. Energy is measured first at time $t_1$ and the outcome is $E_2$. Then total angular momentum is measured at time $t_2 > t_1$ and finally energy is measured again at $t_3 > t_2$. The probability of finding the system in a state with energy $E_2$ after the final measurement is $P/9$. The value of $P$ is ______________ (in integer).- $H$ 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?
A particle has wavefunction
$\psi(x,y,z) = N ze^{-\alpha(x^2+y^2+z^2)}$,
where $N$ is a normalization constant and $\alpha$ is a positive constant. In this state, which one of the following options represents the eigenvalues of $L^2$ and $L_z$ respectively?
Some values of $Y_l^m$ are:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
