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
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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
