# Chap 38: Photons and Matter Waves

## Sections

| Sec | Topic |
|-----|------|
| 38-1 | [The Photon, the Quantum of Light](38-1-the-photon-the-quantum-of-light.ipynb) |
| 38-2 | [The Photoelectric Effect](38-2-the-photoelectric-effect.ipynb) |
| 38-3 | [Photons, Momentum, Compton Scattering, Light Interference](38-3-photons-momentum-compton-scattering-light-interference.ipynb) |
| 38-4 | [The Birth of Quantum Physics](38-4-the-birth-of-quantum-physics.ipynb) |
| 38-5 | [Electrons and Matter Waves](38-5-electrons-and-matter-waves.ipynb) |
| 38-6 | [Schrödinger's Equation](38-6-schrodingers-equation.ipynb) |
| 38-7 | [Heisenberg's Uncertainty Principle](38-7-heisenbergs-uncertainty-principle.ipynb) |
| 38-8 | [Reflection from a Potential Step](38-8-reflection-from-a-potential-step.ipynb) |
| 38-9 | [Tunneling Through a Potential Barrier](38-9-tunneling-through-a-potential-barrier.ipynb) |

## Review & Summary

:::{glossary}
The Photon
  Light is quantized in **photons.** The energy and momentum of a photon of frequency $f$ and wavelength $\lambda$ are

  $$
  E = hf = \frac{hc}{\lambda}, \quad p = \frac{h}{\lambda} = \frac{E}{c}
  $$ (eq-38-photon)

  where $h = 6.63 \times 10^{-34}$ J·s is Planck's constant.

The Photoelectric Effect
  When light strikes a metal surface, electrons can be ejected. **Einstein's equation:**

  $$
  K_{\max} = hf - \Phi
  $$ (eq-38-photoelectric)

  where $K_{\max}$ is the maximum kinetic energy of ejected electrons and $\Phi$ is the **work function** of the metal. The **threshold frequency** below which no electrons are ejected is $f_0 = \Phi/h$.

Compton Scattering
  When an x-ray photon scatters from an electron, the wavelength shift (**Compton shift**) is

  $$
  \Delta\lambda = \frac{h}{m_e c}(1 - \cos\phi)
  $$ (eq-38-compton)

  where $\phi$ is the scattering angle and $h/(m_e c)$ is the Compton wavelength of the electron.

Electrons and Matter Waves
  **de Broglie wavelength:** A particle with momentum $p$ has an associated wavelength $\lambda = h/p = h/(mv)$. Electrons and other particles exhibit wave-like behavior (diffraction, interference).

Schrödinger's Equation
  The wave function $\Psi$ obeys the **Schrödinger equation**, which governs the time evolution of quantum systems. For a free particle, $\Psi \propto e^{i(kx - \omega t)}$ with $E = \hbar\omega$ and $p = \hbar k$, where $\hbar = h/(2\pi)$.

Heisenberg's Uncertainty Principle
  Position and momentum cannot both be known with arbitrary precision:

  $$
  \Delta x \, \Delta p \geq \frac{\hbar}{2}
  $$ (eq-38-uncertainty-xp)

  Similarly, $\Delta E \, \Delta t \geq \hbar/2$ for energy and time.

Reflection and Tunneling
  A matter wave can reflect from a potential step even when the particle's energy exceeds the step height. **Quantum tunneling:** a particle can penetrate a potential barrier of height $U > E$ with a probability that decreases exponentially with barrier width.
:::