Atomic and Molecular Physics

"Periodic Table of Elements.csv" https://gist.github.com/GoodmanSciences/c2dd862cd38f21b0ad36b8f96b4bf1ee

TODO

• Radial
• Angular
• Probable radius
• Degeneracy

Part 1 - One-Electron Atoms

Z

Z = 1 for neutral atoms Z = ion count + 1

Rydberg

Energy Levels

The energy eigenvalues, E, can be determined using the Schrodinger equation $$\hat{H} \Psi = E \Psi$$ Which leads to a set of $E_n$ given by

$$E_n = -\frac{m_e Z^2_{core} e^4}{32 \pi^2 \epsilon^2_0 \hbar^2 n^2}$$

where n = 1, 2, 3, ... is the principle quantum number.

.Figure. 1 Electronic energy levels and Coulomb potential.

Emission

Absorption

Spectrum Generation

Reduced Mass

Radial

$$R_{n, l} = N_{n, l} (\frac{2}{n a_0} r)^l exp(-\frac{1}{n a_0} r) L^{2l+1}_{n-l-1} (\frac{2}{n a_0} r)$$

Where $$N_{n, l} = $$

and $$L $$

Angular Momentum

Part 2 - Multi-Electron Atoms

Electronic Hamiltonian

The part at the end represents the electron-electron repulsion term $V_{rep}$

$$\hat{H}(\vec{r_1},\vec{r_2},\vec{r_3},...,\vec{r_N}) = \sum_{i=1}^N(-\frac{\hbar^2}{2m_e}\nabla_i^2 - \frac{Z_{nucl}e^2}{4 \pi \epsilon_0 r_i}) + \sum_{i=1}^N \sum_{j>1}^N\frac{e^2}{4 \pi \epsilon_0 r_{ij})$$

Independent-Particle Model

$$\hat{H}(\vec{r_1},\vec{r_2},\vec{r_3},...,\vec{r_N}) = \sum_i \hat{h}(\vec{r_i})$$

Central-Field Approximation

Part 3 - Indistinguishable Particles

The Pauli Principle

Spin Wavefunctions

Helium Atom Wavefunctions

Spin Multiplicity

Exhange

Electronic Configurations