Formula Sheet — Tutorial 15: Modern Physics

1. Blackbody Radiation

Wien's Displacement Law

$$\lambda_{max} = \frac{b}{T}$$

or

$$\lambda_{max} T = b$$

Variable Meaning Units
$\lambda_{max}$ Peak wavelength m
$T$ Absolute temperature K
$b$ Wien's displacement constant $2.90 \times 10^{-3} \text{ m K}$

Stefan-Boltzmann Law (Total Power Radiated)

$$P = \sigma A T^4$$

Variable Meaning Units
$P$ Power radiated W
$\sigma$ Stefan-Boltzmann constant $5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$
$A$ Surface area $\text{m}^2$
$T$ Absolute temperature K

Planck's Radiation Law (Spectral Radiance)

$$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}$$

Variable Meaning Units
$B(\lambda, T)$ Spectral radiance $\text{W sr}^{-1} \text{ m}^{-3}$
$k_B$ Boltzmann constant $1.38 \times 10^{-23} \text{ J/K}$

2. Photon Energy & Momentum

Photon Energy

$$E = hf = \frac{hc}{\lambda}$$

Variable Meaning Units
$E$ Photon energy J or eV
$h$ Planck constant $6.626 \times 10^{-34} \text{ J s}$
$f$ Frequency Hz
$\lambda$ Wavelength m
$c$ Speed of light $3.0 \times 10^8 \text{ m/s}$

Photon Momentum

$$p = \frac{E}{c} = \frac{h}{\lambda} = \frac{hf}{c}$$

Variable Meaning Units
$p$ Photon momentum kg m/s

Energy-Momentum Relation for Massless Particles

$$E = pc$$

3. de Broglie Waves

de Broglie Wavelength

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$

Variable Meaning Units
$\lambda$ de Broglie wavelength m
$p$ Momentum kg m/s
$m$ Mass of particle kg
$v$ Velocity of particle m/s

de Broglie Wavelength from Kinetic Energy

For non-relativistic particles:

$$KE = \frac{1}{2}mv^2 \Rightarrow v = \sqrt{\frac{2 KE}{m}}$$

$$\lambda = \frac{h}{m\sqrt{\frac{2 KE}{m}}} = \frac{h}{\sqrt{2m \cdot KE}}$$

de Broglie Wavelength from Accelerating Voltage

For a particle accelerated through potential difference $V$:

$$KE = eV = \frac{1}{2}mv^2 \Rightarrow v = \sqrt{\frac{2eV}{m}}$$

$$\lambda = \frac{h}{\sqrt{2meV}}$$

Variable Meaning Units
$e$ Elementary charge $1.602 \times 10^{-19} \text{ C}$
$V$ Accelerating potential V

Kinetic Energy of a Particle

$$KE = \frac{1}{2}mv^2 = \frac{p^2}{2m}$$

4. Heisenberg Uncertainty Principle

Position-Momentum Uncertainty

$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$

or equivalently:

$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$

Variable Meaning Units
$\Delta x$ Uncertainty in position m
$\Delta p$ Uncertainty in momentum kg m/s
$\hbar$ Reduced Planck constant $1.055 \times 10^{-34} \text{ J s}$

Minimum Uncertainty in Momentum

$$(\Delta p)_{min} = \frac{\hbar}{2\Delta x} = \frac{h}{4\pi \Delta x}$$

Energy-Time Uncertainty

$$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$

Variable Meaning Units
$\Delta E$ Uncertainty in energy J
$\Delta t$ Uncertainty in time s

5. Photoelectric Effect

Einstein's Photoelectric Equation

$$KE_{max} = hf - \phi$$

Variable Meaning Units
$KE_{max}$ Maximum kinetic energy of emitted electrons J or eV
$hf$ Energy of incident photon J or eV
$\phi$ Work function of metal J or eV

Work Function from Threshold Frequency

$$\phi = hf_0$$

Variable Meaning Units
$f_0$ Threshold frequency Hz

Stopping Potential

$$eV_0 = KE_{max} = hf - \phi$$

Variable Meaning Units
$V_0$ Stopping potential V
$e$ Elementary charge $1.602 \times 10^{-19} \text{ C}$

Threshold Wavelength

$$\lambda_0 = \frac{c}{f_0} = \frac{hc}{\phi}$$

Variable Meaning Units
$\lambda_0$ Threshold wavelength m

Photoelectric Equation with Stopping Potential & Wavelength

$$eV_0 = \frac{hc}{\lambda} - \frac{hc}{\lambda_0}$$

Solving for threshold wavelength:

$$\lambda_0 = \frac{hc}{\frac{hc}{\lambda} - eV_0}$$

6. Summary Table of Key Constants

Constant Symbol Value
Planck constant $h$ $6.626 \times 10^{-34} \text{ J s}$
Reduced Planck constant $\hbar$ $1.055 \times 10^{-34} \text{ J s}$
Speed of light $c$ $3.0 \times 10^8 \text{ m/s}$
Wien's displacement constant $b$ $2.90 \times 10^{-3} \text{ m K}$
Stefan-Boltzmann constant $\sigma$ $5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$
Boltzmann constant $k_B$ $1.38 \times 10^{-23} \text{ J/K}$
Elementary charge $e$ $1.602 \times 10^{-19} \text{ C}$
Electron mass $m_e$ $9.11 \times 10^{-31} \text{ kg}$
Proton mass $m_p$ $1.67 \times 10^{-27} \text{ kg}$
1 eV in joules $1 \text{ eV}$ $1.602 \times 10^{-19} \text{ J}$

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