Modern Physics

Quantum theory was developed to explain results such as atomic spectra, blackbody radiation and the photoelectric effect. Light can be modeled as a wave or as photons: massless, neutral particles with energy E = hf that travel at c in a vacuum. Particles like electrons also act as waves, with a de Broglie wavelength $\lambda = \frac{h}{p}$, so a smaller momentum means a longer wavelength, and electrons sent through a double slit make an interference pattern.

An atom has a tiny positive nucleus of protons and neutrons, which holds almost all of its mass, surrounded by electrons. The number of protons identifies the element, and the total number of protons and neutrons identifies the isotope. In the Bohr model, the electric force from the nucleus keeps the electron in a circular orbit, and only orbits whose circumference fits a whole number of the electron's de Broglie wavelengths are allowed, which is why the atom has only certain energy levels.

An atom can absorb or emit a photon only when the photon's energy exactly matches the difference between two of its energy levels, so each transition gives one frequency and one wavelength of light. Because every element has its own set of levels, its emission or absorption spectrum works like a fingerprint you can use to identify it. On an energy-level diagram, the ground state is the lowest level, and removing the electron from the ground state (ionization) takes the most energy.

Every object turns some of its thermal energy into electromagnetic radiation. An ideal blackbody absorbs all the radiation that hits it and gives off a continuous spectrum that depends only on its temperature, which classical physics couldn't explain until Planck assumed light's energy is quantized. As temperature goes up, the peak wavelength gets shorter (Wien's law: $\lambda_{\max}T$ is a constant) and the power emitted grows with surface area and with T⁴ (the Stefan-Boltzmann law).

In the photoelectric effect, light shining on a metal knocks electrons out, but only if its frequency is at or above a threshold frequency, no matter how bright the light is. The fastest ejected electrons have $K_{\text{max}} = hf - \phi$, where the work function φ is the least energy needed to free an electron (the exam gives you its value). Brighter light ejects more electrons but doesn't make them faster, which is evidence that light comes in photons; in experiments, a stopping voltage that just shuts off the current measures $K_{\text{max}}$.

In Compton scattering, a photon (usually an X-ray) collides with a free electron and comes out with less energy and a longer wavelength. You can explain it by treating the photon as a particle with momentum $p = \frac{h}{\lambda}$ and applying conservation of energy and momentum in two dimensions, and the bigger the photon's change in direction, the bigger its change in wavelength. Like the photoelectric effect, it's evidence that light comes in photons.

The strong force holds protons and neutrons together in the nucleus. Nuclear reactions conserve nucleon number, charge, energy and momentum, and mass can turn into energy through E = mc², released as kinetic energy of the products or as photons. Fusion joins small nuclei into a bigger one, and fission splits a big nucleus into smaller ones. Radioactive decay happens on its own: a nucleus turns into one or more different nuclei, or stays the same nucleus and drops to a lower energy level (a fall 2026 clarification). Decay is random for any single nucleus, but a sample has a half-life, the time for half its nuclei to decay, related to the decay constant by $t_{1/2} = \frac{\ln 2}{\lambda}$.

Unstable nuclei decay in four ways you need to know. In alpha decay a nucleus ejects a helium-4 nucleus; in beta-minus decay a neutron becomes a proton and emits an electron and an antineutrino; in beta-plus decay a proton becomes a neutron and emits a positron and a neutrino; and in gamma decay an excited nucleus drops to a lower energy state by emitting a photon. Every decay conserves nucleon number, charge and lepton number, which is how you balance decay equations.