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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
Quantum Mechanics Wavefunction Analysis
The given wavefunction is $\psi(x,y,z) = N ze^{-\alpha(x^2+y^2+z^2)}$. We need to find the eigenvalues of the angular momentum operators $L^2$ and $L_z$. These operators act on the angular part of the wavefunction in spherical coordinates.
Convert to Spherical Coordinates
First, convert the wavefunction to spherical coordinates $(r, \theta, \phi)$. We know that $z = r \cos\theta$ and $x^2+y^2+z^2 = r^2$. Substituting these, the wavefunction becomes:
$\psi(r, \theta, \phi) = N (r \cos\theta) e^{-\alpha r^2}$
We can separate this into a radial part $R(r)$ and an angular part $Y(\theta, \phi)$: $R(r) = N r e^{-\alpha r^2}$ $Y(\theta, \phi) = \cos\theta$
Identify Angular Momentum Quantum Numbers
The eigenvalues of $L^2$ and $L_z$ depend on the angular momentum quantum numbers $l$ and $m$, respectively, which characterize the angular part of the wavefunction (Spherical Harmonics, $Y_l^m$).
The eigenvalue equation for $L_z$ is $L_z Y_l^m = m\hbar Y_l^m$. The eigenvalue equation for $L^2$ is $L^2 Y_l^m = l(l+1)\hbar^2 Y_l^m$.
We are given some Spherical Harmonics:
- $Y_0^0 = \sqrt{\frac{1}{4\pi}}$
- $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$
- $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
Our angular part is $Y(\theta, \phi) = \cos\theta$. We can express $\cos\theta$ in terms of the given $Y_1^0$:
$\cos\theta = \sqrt{\frac{4\pi}{3}} Y_1^0(\theta, \phi)$
Thus, the angular part of our wavefunction is proportional to $Y_1^0$. This means the quantum numbers are $l=1$ and $m=0$.
Calculate Eigenvalues
Now, we calculate the eigenvalues using $l=1$ and $m=0$:
- $L^2$ Eigenvalue: $l(l+1)\hbar^2 = 1(1+1)\hbar^2 = 1(2)\hbar^2 = 2\hbar^2$.
- $L_z$ Eigenvalue: $m\hbar = 0 \cdot \hbar = 0$.
The eigenvalues for $L^2$ and $L_z$ are $2\hbar^2$ and $0$, respectively.
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?
- The spin $ \vec{S}$ and orbital angular momentum $ \vec{L}$ of an atom precess about $ \vec{J}$, the total angular momentum. $ \vec{J}$ precesses about an axis fixed by a magnetic field $ \vec{B}_1 = 2B_0 \hat{z}$, where $B_0$ is a constant. Now the magnetic field is changed to $ \vec{B}_2 = B_0( \hat{x} + \sqrt{2} \hat{y} + \hat{z})$. Given the orbital angular momentum quantum number $l = 2$ and spin quantum number $s = 1/2$, $ \theta$ is the angle between $ \vec{B}_1$ and $ \vec{J}$ for the largest possible values of total angular quantum number $j$ and its $z$-component $j_z$. The value of $ \theta$ (in degree, rounded off to the nearest integer) is ________
