Orbital Angular Momentum Uncertainty: Hydrogen Atom State
The question asks for the value of $\alpha$, where the uncertainty squared in the y-component of orbital angular momentum, $(\Delta L_y)^2$, is given by $\alpha\hbar^2$ for an electron in a hydrogen atom state $|n=3, l=2, m=-2\rangle$.
Calculating $(\Delta L_y)^2$
The uncertainty squared is defined as $(\Delta L_y)^2 = \langle L_y^2 \rangle - \langle L_y \rangle^2$.
Step 1: Expectation Value $\langle L_y \rangle$
For a state with a definite magnetic quantum number $m$, the expectation value $\langle L_y \rangle$ is zero.
$ \langle L_y \rangle = \langle n, l, m | \hat{L}_y | n, l, m \rangle = 0 $
Step 2: Expectation Value $\langle L_y^2 \rangle$
The expectation value of $L_y^2$ for the state $|n, l, m\rangle$ is given by:
$ \langle L_y^2 \rangle = \frac{1}{2} [l(l+1) - m^2] \hbar^2 $
Using the given quantum numbers $l=2$ and $m=-2$:
- Substitute values: $ \langle L_y^2 \rangle = \frac{1}{2} [2(2+1) - (-2)^2] \hbar^2 $
- Simplify: $ \langle L_y^2 \rangle = \frac{1}{2} [6 - 4] \hbar^2 = \frac{1}{2} [2] \hbar^2 = \hbar^2 $
Step 3: Calculate Variance $(\Delta L_y)^2$
Now, calculate the variance using the expectation values found:
$ (\Delta L_y)^2 = \langle L_y^2 \rangle - \langle L_y \rangle^2 = \hbar^2 - 0^2 = \hbar^2 $
Step 4: Determine $\alpha$
We are given $(\Delta L_y)^2 = \alpha\hbar^2$. Comparing this with our result:
$ \alpha\hbar^2 = \hbar^2 $
Solving for $\alpha$ yields:
$ \alpha = 1 $
Final Result
The value of $\alpha$ is 1. This confirms the range provided (between 1 and 1).
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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
