Formula Sheet — Modern Physics (Wave-Particle Duality)
Comprehensive formula reference extracted from Modern Physics — Wave-Particle Duality. All formulas use standard SI units unless otherwise noted.
1. Photon Energy & Momentum
Photon Energy
$$E = hf = \hbar\omega = \frac{hc}{\lambda}$$
| Variable | Meaning | Units |
|---|---|---|
| $E$ | Photon energy | J (or eV) |
| $h$ | Planck constant ($h = 6.626 \times 10^{-34}\ \text{J}\cdot\text{s}$) | J·s |
| $f$ | Photon frequency | Hz |
| $\hbar$ | Reduced Planck constant | J·s |
| $\omega$ | Angular frequency ($\omega = 2\pi f$) | rad/s |
| $c$ | Speed of light | m/s |
| $\lambda$ | Wavelength | m |
Photon Momentum
$$p = \frac{h}{\lambda} = \hbar k$$
| Variable | Meaning | Units |
|---|---|---|
| $p$ | Photon momentum | kg·m/s |
| $h$ | Planck constant | J·s |
| $\lambda$ | Wavelength | m |
| $\hbar$ | Reduced Planck constant | J·s |
| $k$ | Wave number ($k = \frac{2\pi}{\lambda}$) | rad/m |
2. Photoelectric Effect
Einstein's Photoelectric Equation
$$K_{\max} = hf - \phi = eV_0$$
| Variable | Meaning | Units |
|---|---|---|
| $K_{\max}$ | Maximum kinetic energy of emitted electrons | J (or eV) |
| $h$ | Planck constant | J·s |
| $f$ | Incident light frequency | Hz |
| $\phi$ | Work function of the material | J (or eV) |
| $e$ | Elementary charge | C |
| $V_0$ | Stopping potential | V |
Threshold frequency: $f_0 = \frac{\phi}{h}$ (minimum frequency to eject electrons)
Threshold wavelength: $\lambda_{\max} = \frac{hc}{\phi}$ (maximum wavelength to eject electrons)
3. Compton Effect
Compton Wavelength Shift
$$\lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)$$
| Variable | Meaning | Units |
|---|---|---|
| $\lambda'$ | Scattered photon wavelength | m |
| $\lambda$ | Incident photon wavelength | m |
| $h$ | Planck constant | J·s |
| $m_e$ | Electron rest mass | kg |
| $c$ | Speed of light | m/s |
| $\theta$ | Photon scattering angle | rad (or °) |
Compton wavelength of the electron: $$\lambda_C = \frac{h}{m_e c} \approx 2.43 \times 10^{-12}\ \text{m}$$
4. De Broglie Hypothesis (Matter Waves)
De Broglie Wavelength
$$\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2mK}}$$
| Variable | Meaning | Units |
|---|---|---|
| $\lambda$ | De Broglie wavelength | m |
| $h$ | Planck constant | J·s |
| $p$ | Particle momentum | kg·m/s |
| $m$ | Particle mass | kg |
| $v$ | Particle speed | m/s |
| $K$ | Kinetic energy | J |
For a particle accelerated through potential $V$: $$\lambda = \frac{h}{\sqrt{2meV}}$$
5. Heisenberg Uncertainty Principle
Position–Momentum Uncertainty
$$\Delta x , \Delta p \geq \frac{\hbar}{2}$$
| Variable | Meaning | Units |
|---|---|---|
| $\Delta x$ | Uncertainty in position | m |
| $\Delta p$ | Uncertainty in momentum | kg·m/s |
| $\hbar$ | Reduced Planck constant | J·s |
Energy–Time Uncertainty
$$\Delta E , \Delta t \geq \frac{\hbar}{2}$$
| Variable | Meaning | Units |
|---|---|---|
| $\Delta E$ | Uncertainty in energy | J (or eV) |
| $\Delta t$ | Uncertainty in time | s |
| $\hbar$ | Reduced Planck constant | J·s |
6. Blackbody Radiation
Planck's Law (Spectral Energy Density)
$$u(\lambda, T) = \frac{8\pi hc}{\lambda^5}\frac{1}{e^{hc/\lambda k_B T} - 1}$$
| Variable | Meaning | Units |
|---|---|---|
| $u(\lambda, T)$ | Spectral energy density (energy per unit volume per unit wavelength) | J·m⁻⁴ |
| $h$ | Planck constant | J·s |
| $c$ | Speed of light | m/s |
| $\lambda$ | Wavelength | m |
| $k_B$ | Boltzmann constant ($k_B = 1.381 \times 10^{-23}\ \text{J/K}$) | J/K |
| $T$ | Absolute temperature | K |
Wien's Displacement Law
$$\lambda_{\max} = \frac{b}{T}$$
| Variable | Meaning | Units |
|---|---|---|
| $\lambda_{\max}$ | Wavelength at maximum intensity | m |
| $b$ | Wien's displacement constant ($b = 2.90 \times 10^{-3}\ \text{m}\cdot\text{K}$) | m·K |
| $T$ | Absolute temperature | K |
Stefan–Boltzmann Law
$$\frac{P}{A} = \sigma T^4$$
| Variable | Meaning | Units |
|---|---|---|
| $P$ | Total radiated power | W |
| $A$ | Surface area of the blackbody | m² |
| $\sigma$ | Stefan–Boltzmann constant ($\sigma = 5.67 \times 10^{-8}\ \text{W}\cdot\text{m}^{-2}\cdot\text{K}^{-4}$) | W·m⁻²·K⁻⁴ |
| $T$ | Absolute temperature | K |
7. Radiation Properties & Absorptivity
Absorptivity Identity
$$\alpha_\nu + \rho_\nu + \tau_\nu = 1$$
| Variable | Meaning | Units |
|---|---|---|
| $\alpha_\nu$ | Absorptivity (fraction absorbed) | dimensionless |
| $\rho_\nu$ | Reflectivity (fraction reflected) | dimensionless |
| $\tau_\nu$ | Transmissivity (fraction transmitted) | dimensionless |
For a perfect blackbody: $\alpha_\nu = 1$ and $\rho_\nu = \tau_\nu = 0$.
Wave–Particle Comparison Summary
| Property | Wave | Particle |
|---|---|---|
| Energy relation | $E \propto \text{Intensity}$ | $E = hf$ (discrete packets) |
| Interference | Yes | No |
| Localization | Spread out | Localized collisions |
8. Key Constants
| Constant | Symbol | Value |
|---|---|---|
| Planck constant | $h$ | $6.626 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| Reduced Planck constant | $\hbar$ | $1.055 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| Speed of light | $c$ | $3.00 \times 10^8\ \text{m/s}$ |
| Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}\ \text{J/K}$ |
| Stefan–Boltzmann constant | $\sigma$ | $5.67 \times 10^{-8}\ \text{W}\cdot\text{m}^{-2}\cdot\text{K}^{-4}$ |
| Wien's displacement constant | $b$ | $2.90 \times 10^{-3}\ \text{m}\cdot\text{K}$ |
| Electron mass | $m_e$ | $9.109 \times 10^{-31}\ \text{kg}$ |
| Electron charge | $e$ | $1.602 \times 10^{-19}\ \text{C}$ |
| Compton wavelength (electron) | $\lambda_C$ | $2.43 \times 10^{-12}\ \text{m}$ |