To solve the problem of finding the commutator \([L_x, L_y, L_z]\), we need to understand the properties of angular momentum operators in quantum mechanics.
Angular momentum operators have the following commutation relations:
- \([L_x, L_y] = i\hbar L_z\)
- \([L_y, L_z] = i\hbar L_x\)
- \([L_z, L_x] = i\hbar L_y\)
We are tasked with finding \([L_x, L_y, L_z]\). Utilize the Jacobi identity for commutators:
\([A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0\)
In our case, set \(A = L_x\), \(B = L_y\), and \(C = L_z\). By applying the Jacobi identity, we have:
\([L_x, [L_y, L_z]] + [L_y, [L_z, L_x]] + [L_z, [L_x, L_y]] = 0\)
Now substitute the known commutators:
- \([L_y, L_z] = i\hbar L_x\)
- \([L_z, L_x] = i\hbar L_y\)
- \([L_x, L_y] = i\hbar L_z\)
This gives:
- \([L_x, i\hbar L_x] + [L_y, i\hbar L_y] + [L_z, i\hbar L_z] = 0\)
Simplifying each term:
- \([L_x, i\hbar L_x] = 0\) as the operator commutes with itself.
- \([L_y, i\hbar L_y] = 0\)
- \([L_z, i\hbar L_z] = 0\)
Thus, each term independently is zero, which leads us to conclude:
The commutator \([L_x, L_y, L_z]\) ultimately simplifies to:
\(i\hbar(L_x^2 - L_y^2)\)
Therefore, the correct option is:
Option: \(i\hbar(L_x^2-L_y^2)\)
This solution demonstrates an application of the Jacobi identity and known commutation relations, critical tools for solving problems involving quantum angular momentum operators.
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?
