Question
Which one of the following commutation relations is NOT CORRECT ? Here, symbols have their usual meanings.
The correct answer is
$[L_z, L_-] = \hbar L_-$
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: $[Lz, L-] = -ħ L-$.
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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?
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