• The energy of a light wave comes in discrete packets called light quanta or photons.

• For light of frequency $f$, the energy of each photon is $E=hf$, where $h$ is Planck’s constant.

• For a light wave that delivers power $P$, photons arrive at rate $R$ such that $P=Rhf$.

• Photons are “particle-like” but are not classical particles.

• The de Broglie wavelength of a “particle” of mass $m$ is $\lambda =h\text{/}mv$.

• The wave-like nature of matter is seen in the interference patterns of electrons, neutrons, and entire atoms.

• When a particle is confined, it sets up a de Broglie standing wave. The fact that standing waves have only certain allowed wavelengths leads to the conclusion that a confined particle has only certain allowed energies. That is, energy is quantized.

  

• Light consists of quanta of energy $E=hf$ and momentum $p=h/\lambda$.

• Quanta are emitted and absorbed on an all-or-nothing basis.

• When a light quantum is absorbed, it delivers all its energy to one electron.

• An atom can exist in only certain stationary states. The allowed energies are quantized. State $n$ has energy $E_n$.

• An atom can transition from one stationary state to another by emitting or absorbing a photon with $E_{\text{photon}}=hf=\text{Δ}{E}_{\text{atom}}$.

• Atoms can be excited in inelastic collisions.

• Atoms seek the $n=1$ ground state. Most atoms, most of the time, are in the ground state.

  

Light can eject electrons from a metal only if $f\ge f_0=E_0\text{/}h$, where $E_0$ is the metal’s work function.

The stopping potential that stops even the fastest electrons is

An integer number of de Broglie wavelengths must fit around the circumference of the electron’s orbit: $2\pi r=n\lambda$.

This leads to energy quantization with

where $a_{\text{B}}=0.0529\text{nm}$ is the Bohr radius.

A particle confined to a one-dimensional box of length $L$ sets up de Broglie standing waves. The allowed energies are

When x rays scatter from atoms at angle $\theta$, their wavelength increases by