Wavefunction Angular Part Analysis
The angular part of the wavefunction for an electron in a hydrogen atom is given by spherical harmonics, $Y_l^m(\theta, \phi)$. This part depends on the azimuthal quantum number ($l$) and the magnetic quantum number ($m$).
The general form is $Y_l^m(\theta, \phi) \propto P_l^{|m|}(\cos \theta) e^{im\phi}$, where $P_l^{|m|}$ represents the associated Legendre polynomial.
Identifying Magnetic Quantum Number ($m$)
The term $e^{im\phi}$ in the wavefunction directly corresponds to the magnetic quantum number, $m$.
Given angular part: $\propto \sin^2 \theta \cos \theta e^{2i\phi}$.
Comparing the exponential term $e^{2i\phi}$ with the general form $e^{im\phi}$, we identify $m = 2$.
Determining Azimuthal Quantum Number ($l$)
The $\theta$ dependence of the wavefunction, along with the value of $m$, determines the azimuthal quantum number, $l$. We match the given $\theta$ dependence ($\sin^2 \theta \cos \theta$) with known spherical harmonics.
Consider the spherical harmonic $Y_3^2(\theta, \phi)$. Its form is known to be proportional to $\sin^2 \theta \cos \theta e^{2i\phi}$.
This matches the provided angular part of the wavefunction.
Therefore, by comparing $Y_3^2(\theta, \phi)$ with the general form $Y_l^m(\theta, \phi)$, we conclude that $l=3$ and $m=2$.
Conclusion
The azimuthal quantum number ($l$) is 3.
The magnetic quantum number ($m$) is 2.
Thus, the values are $l=3$ and $m=2$.
Important Questions from Hydrogen Atom
- The experimental ionization energies of hydrogen and helium atoms in their ground states are, respectively, 13.6 eV and 24.6 eV. The ground state energy of helium atom, in eV, is
- $ψ = N r(6 – Zr)e^{-Zr/3} cos θ$, is a proposed hydrogenic wavefunction, where $Z$ = Atomic number, $r$ = radial distance from the nucleus, $θ$ = azimuthal angle, $N$ is a constant. The INCORRECT statement about $ψ$ is
- The positions of two atoms in spherical polar coordinates ($r, \theta, \Phi$) are ($1, \frac{\pi}{2}, \frac{\pi}{2}$) and ($1, \frac{\pi}{4}, \frac{3\pi}{2}$), where the distance is in Å and the angles are in radian. The interatomic distance (in Å) is ______ (rounded off to two decimal places).
