Angular Momentum Commutation Relations Analysis
This question requires identifying the incorrect commutation relation among standard angular momentum operators in quantum mechanics. The commutation relation $[A, B]$ is defined as $AB - BA$. We will examine each option based on the known properties of angular momentum operators.
Standard Commutation Relations
The standard commutation relations for angular momentum operators are:
- $[Lx, Ly] = iħ Lz$
- $[Ly, Lz] = iħ Lx$
- $[Lz, Lx] = iħ Ly$
- $[L^2, Li] = 0$, where $i = x, y, z$
- $[Lz, L+] = ħ L+$, where $L+ = Lx + iLy$ (raising operator)
- $[Lz, L-] = -ħ L-$, where $L- = Lx - iLy$ (lowering operator)
Evaluating the Options
Let's check each provided option against the standard relations:
-
Option 1: $[L^2, Lz] = 0$
This relation states that the total angular momentum squared operator, $L^2$, commutes with the z-component, $L_z$. This is a standard and correct property of angular momentum operators. -
Option 2: $[Lx, Ly] = iħ Lz$
This is the fundamental commutation relation between the x and y components of angular momentum. It is correct. -
Option 3: $[Lz, L+] = ħ L+$
This relation involves the z-component and the angular momentum raising operator $L_+$. This is a standard and correct commutation relation. -
Option 4: $[Lz, L-] = ħ L-$
This option relates the z-component and the angular momentum lowering operator $L_-$. However, the standard, correct commutation relation is $[Lz, L-] = -ħ L-$. The sign is different. Therefore, this relation is NOT CORRECT.
Conclusion
Based on the standard commutation relations in quantum mechanics, the relation $[Lz, L-] = ħ L-$ is incorrect. The correct relation has a negative sign: $[Lz, L-] = -ħ L-$.
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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
