Angular Momentum Ladder Operator Expectation Value
This solution determines the expectation value of the operator $(L_+L_- + L_-L_+)$ in the quantum state $ |l = 1, m = 1\rangle $.
Operator Identity for (L_+L_- + L_-L_+)
The operator $(L_+L_- + L_-L_+)$ is simplified using the identity relating it to the total angular momentum squared operator $L2$ and the z-component operator $Lz2$:
$ L_+L_- + L_-L_+ = 2(L^2 - L_z^2) $
Expectation Value Calculation in State $|l=1, m=1\rangle$
We calculate the expectation value:
$ \langle l=1, m=1 | (L_+L_- + L_-L_+) | l=1, m=1 \rangle = \langle l=1, m=1 | 2(L^2 - L_z^2) | l=1, m=1 \rangle $
$ = 2 \left( \langle l=1, m=1 | L^2 | l=1, m=1 \rangle - \langle l=1, m=1 | L_z^2 | l=1, m=1 \rangle \right) $
The eigenvalues for $L2$ and $Lz2$ in a state $|l, m\rangle$ are:
- $L2$ eigenvalue: $l(l+1)\hbar^2$
- $Lz2$ eigenvalue: $m^2\hbar^2$
For the state $|l = 1, m = 1\rangle$:
- Expectation value of $L2$: $\langle L^2 \rangle = 1(1+1)\hbar^2 = 2\hbar^2$
- Expectation value of $Lz2$: $\langle L_z^2 \rangle = 1^2\hbar^2 = 1\hbar^2$
Substituting these values:
$ \text{Expectation Value} = 2 (2\hbar^2 - 1\hbar^2) = 2 (1\hbar^2) = 2\hbar^2 $
The expectation value is $2$ in units of $ \hbar^2 $.
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?
