Hydrogen-like Atomic Orbitals and Schrödinger Equation – Exam Prep Notes
The Bohr Model of Hydrogen
Treats hydrogen with electrons in stationary orbits with quantized radii (rn = a0 n^2) and energies (E_n = - \frac{13.6\ \text{eV}}{n^2}).
Explains spectral lines via the Rydberg formula (\frac{1}{\lambda} = RH\left( \frac{1}{n1^2} - \frac{1}{n2^2} \right)), categorizing transitions into series like Lyman (n1 = 1), Balmer (n_1 = 2), etc., which arise from quantized energy differences.
The Schrödinger Equation and Atomic Orbitals
Provides spatial probability densities (|\psi(r,\theta,\phi)|^2) for electron positions, reflecting a quantum mechanical view.
Solutions (wavefunctions) are labeled by four quantum numbers:
Principal quantum number (n): 1,2,3,\dots, defines energy and size.
Angular momentum quantum number (l): 0,1,2,\dots,(n-1), defines orbital shape (s, p, d, f).
Magnetic quantum number (m): -l,\dots,l, defines orbital orientation.
Spin quantum number (m_s): +\frac{1}{2}, -\frac{1}{2}, describes electron spin.
Wavefunctions separate into radial (R{n,l}(r)) and angular (Yl^m(\theta,\phi)) parts: \psi{n,l,m}(r,\theta,\phi) = R{n,l}(r)\,Y_l^m(\theta,\phi).
Shapes and Radial Probability Distributions
Radial probability density (P{\text{radial}}(r) = 4\pi r^2 |R{n,l}(r)|^2 |Y_l^m(\theta,\phi)|^2) gives the probability of finding an electron at distance r.
Orbitals have nodes (regions of zero probability):
Total nodes: n-1
Radial nodes: n-l-1
s orbitals (l=0) have non-zero density at the nucleus (high penetration), while higher l orbitals have nodes near the nucleus.
Increasing n for a given orbital type makes the orbital larger and increases the number of radial nodes.
The Hydrogenic Orbitals and Energy Degeneracy
In a pure hydrogen-like atom, all orbitals with the same n are degenerate (have the same energy).
In multi-electron atoms, this degeneracy is lifted by electron-electron interactions, making energy dependent on both n and l.