Angular Momentum Calculation
This solution computes a specific angular momentum matrix element involving the raising operator.
Operator Identification
The operator $(L_x+iL_y)$ represents the angular momentum raising operator, commonly denoted as $L_{+}$.
Ladder Operator Action
The effect of the raising operator $L_{+}$ on an angular momentum eigenstate $|l,m\rangle$ is defined by the relation:
$L_{+} |l,m\rangle = \sqrt{l(l+1) - m(m+1)} \hbar |l, m+1\rangle$
Matrix Element Calculation
The goal is to calculate $\langle l,0 | L_{+} | l,-1 \rangle$. First, apply the operator $L_{+}$ to the initial state $|l,-1\rangle$. Here, the azimuthal quantum number is $m=-1$. $L_{+} |l,-1\rangle = \sqrt{l(l+1) - (-1)(-1+1)} \hbar |l, -1+1\rangle$ $L_{+} |l,-1\rangle = \sqrt{l(l+1) - (-1)(0)} \hbar |l, 0\rangle$ $L_{+} |l,-1\rangle = \sqrt{l(l+1)} \hbar |l, 0\rangle$ Next, compute the matrix element:
$\langle l,0 | L_{+} | l,-1 \rangle = \langle l,0 | \left( \sqrt{l(l+1)} \hbar |l, 0\rangle \right)$
Since the states $|l,m\rangle$ are normalized, $\langle l,0 | l,0 \rangle = 1$. Thus:
$\langle l,0 | L_{+} | l,-1 \rangle = \sqrt{l(l+1)} \hbar \langle l,0 | l,0 \rangle = \sqrt{l(l+1)} \hbar \times 1 = \sqrt{l(l+1)} \hbar$
Result Verification
The calculated matrix element is $\sqrt{l(l+1)} \hbar$. One of the options is $\sqrt{2}\hbar$. This implies that $l(l+1)=2$. Solving the quadratic equation $l^2 + l - 2 = 0$ yields $(l+2)(l-1)=0$. As the angular momentum quantum number $l$ must be non-negative ($l \ge 0$), the solution is $l=1$.
Assuming $l=1$, the matrix element becomes:
$\sqrt{1(1+1)} \hbar = \sqrt{2} \hbar$
This matches option C.
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?
