Formula Sheet — Atomic Physics
Comprehensive formula reference extracted from Atomic Physics. All formulas use standard SI units unless otherwise noted.
1. Bohr Model & Atomic Structure
Bohr Radius (Orbit Radius)
$$r_n = n^2 a_0 = (5.29 \times 10^{-11}\ \text{m}),n^2$$
| Variable | Meaning | Units |
|---|---|---|
| $r_n$ | Radius of the $n$-th Bohr orbit | m |
| $n$ | Principal quantum number ($n = 1, 2, 3, \dots$) | dimensionless |
| $a_0$ | Bohr radius ($a_0 = 5.29 \times 10^{-11}\ \text{m}$) | m |
Hydrogen Energy Levels
$$E_n = -\frac{13.6}{n^2}\ \text{eV}$$
| Variable | Meaning | Units |
|---|---|---|
| $E_n$ | Energy of the $n$-th level | eV (or J) |
| $n$ | Principal quantum number | dimensionless |
| $13.6$ | Ionization energy of hydrogen ground state | eV |
Quantized Angular Momentum
$$L = n\hbar = r_n m v_n$$
| Variable | Meaning | Units |
|---|---|---|
| $L$ | Orbital angular momentum | J·s |
| $n$ | Principal quantum number | dimensionless |
| $\hbar$ | Reduced Planck constant | J·s |
| $r_n$ | Bohr orbit radius | m |
| $m$ | Electron mass | kg |
| $v_n$ | Electron orbital speed in $n$-th orbit | m/s |
Reduced Planck Constant
$$\hbar = \frac{h}{2\pi} \approx 1.06 \times 10^{-34}\ \text{J}\cdot\text{s}$$
| Variable | Meaning | Units |
|---|---|---|
| $\hbar$ | Reduced Planck constant | J·s |
| $h$ | Planck constant ($h = 6.626 \times 10^{-34}\ \text{J}\cdot\text{s}$) | J·s |
Reduced Mass (Two-Body System)
$$\mu = \frac{m_e M}{m_e + M}$$
| Variable | Meaning | Units |
|---|---|---|
| $\mu$ | Reduced mass | kg |
| $m_e$ | Electron mass | kg |
| $M$ | Nuclear mass | kg |
2. Atomic Spectra
Rydberg Formula (Hydrogen Spectral Lines)
$$\frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$
| Variable | Meaning | Units |
|---|---|---|
| $\lambda$ | Wavelength of emitted/absorbed photon | m |
| $R_H$ | Rydberg constant ($R_H \approx 1.097 \times 10^7\ \text{m}^{-1}$) | m⁻¹ |
| $n_f$ | Final principal quantum number | dimensionless |
| $n_i$ | Initial principal quantum number ($n_i > n_f$ for emission) | dimensionless |
Spectral Series:
- Lyman series: $n_f = 1$ (UV region)
- Balmer series: $n_f = 2$ (visible region)
- Paschen series: $n_f = 3$ (IR region)
Photon Energy from Transition
$$hf = E_i - E_f$$
| Variable | Meaning | Units |
|---|---|---|
| $h$ | Planck constant | J·s |
| $f$ | Photon frequency | Hz |
| $E_i$ | Initial energy level | J (or eV) |
| $E_f$ | Final energy level | J (or eV) |
Equivalent forms: $$E_{\text{photon}} = hf = \frac{hc}{\lambda} = E_i - E_f$$
Ionization Energy (Hydrogen)
$$E_{\text{ionization}} = 13.6\ \text{eV}$$
Energy required to remove an electron from the ground state ($n = 1$) to infinity ($n = \infty$).
3. Quantum Numbers & Electron Configuration
Quantum Numbers
| Symbol | Name | Allowed Values | Physical Meaning |
|---|---|---|---|
| $n$ | Principal | $1, 2, 3, \dots$ | Shell (K, L, M, N…), energy level |
| $l$ | Orbital (azimuthal) | $0, 1, 2, \dots, (n-1)$ | Subshell shape (s, p, d, f…) |
| $m_l$ | Magnetic | $-l, \dots, 0, \dots, +l$ | Orbital orientation in space |
| $m_s$ | Spin | $+\frac{1}{2}, -\frac{1}{2}$ | Electron spin direction |
Electron Shell Correspondence
$$\text{Shell } n \rightarrow \text{K, L, M, N, \dots for } n = 1, 2, 3, 4, \dots$$
4. Radiation Processes
Stimulated Absorption
Electron absorbs a photon and transitions to a higher energy state. $$E_{\text{photon}} = E_{\text{higher}} - E_{\text{lower}}$$
Spontaneous Emission
Excited electron randomly emits a photon and decays to a lower state (typical lifetime $\sim 10^{-8}$ s). $$E_{\text{photon}} = E_{\text{excited}} - E_{\text{ground}}$$
Stimulated Emission
Incident photon triggers emission of an identical photon (same frequency, phase, polarization, direction). $$E_{\text{photon}} = E_{\text{excited}} - E_{\text{ground}}$$
LASER condition: Population inversion + optical feedback.
5. Key Constants
| Constant | Symbol | Value |
|---|---|---|
| Planck constant | $h$ | $6.626 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| Reduced Planck constant | $\hbar$ | $1.055 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| Bohr radius | $a_0$ | $5.29 \times 10^{-11}\ \text{m}$ |
| Rydberg constant | $R_H$ | $1.097 \times 10^7\ \text{m}^{-1}$ |
| Electron mass | $m_e$ | $9.109 \times 10^{-31}\ \text{kg}$ |
| Proton mass | $m_p$ | $1.673 \times 10^{-27}\ \text{kg}$ |
| Speed of light | $c$ | $3.00 \times 10^8\ \text{m/s}$ |
| Elementary charge | $e$ | $1.602 \times 10^{-19}\ \text{C}$ |
| Hydrogen ionization energy | $E_{\text{ion}}$ | $13.6\ \text{eV} = 2.18 \times 10^{-18}\ \text{J}$ |