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