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 ($r<em>n = a</em>0 n^2$) and energies ($E_n = - \frac{13.6\ \text{eV}}{n^2}$).

• Explains spectral lines via the Rydberg formula ($\frac{1}{\lambda} = R<em>H\left( \frac{1}{n</em>1^2} - \frac{1}{n<em>2^2} \right)$), categorizing transitions into series like Lyman ($n</em>1 = 1$), 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:

  1. Principal quantum number ($n$): $1,2,3,\dots,$ defines energy and size.

  2. Angular momentum quantum number ($l$): $0,1,2,\dots,(n-1),$ defines orbital shape (s, p, d, f).

  3. Magnetic quantum number ($m$): $-l,\dots,l,$ defines orbital orientation.

  4. Spin quantum number ($m_s$): $+\frac{1}{2}, -\frac{1}{2},$ describes electron spin.

• Wavefunctions separate into radial ($R<em>{n,l}(r)$) and angular ($Y</em>l^m(\theta,\phi)$) parts: $\psi<em>{n,l,m}(r,\theta,\phi) = R</em>{n,l}(r)\,Y_l^m(\theta,\phi)$.

Shapes and Radial Probability Distributions

• Radial probability density ($P<em>{\text{radial}}(r) = 4\pi r^2 |R</em>{n,l}(r)|^2 |Y_l^m(\theta,\phi)|^2$) 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 $l$.