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