De Broglie waves and electron diffraction: Electrons and other particles have wave properties. A particle’s wavelength depends on its momentum in the same way as for photons: , . A non-relativistic electron accelerated from rest through a potential difference has a wavelength . Electron microscopes use the very small wavelengths of fast-moving electrons to make images with resolution thousands of times finer than is possible with visible light.
The nuclear atom: The Rutherford scattering experiments show that most of an atom’s mass and all of its positive charge are concentrated in a tiny, dense nucleus at the center of the atom.
Atomic line spectra and energy levels: The energies of atoms are quantized: They can have only certain definite values, called energy levels. When an atom makes a transition from an energy level to a lower level , it emits a photon of energy : . The same photon can be absorbed by an atom in the lower energy level, which excites the atom to the upper level.
The Bohr model: In the Bohr model of the hydrogen atom, the permitted values of angular momentum are integral multiples of : , . The integer multiplier is called the principal quantum number for the level. The orbital radii are proportional to : , . The energy levels of the hydrogen atoms are given by , , where is the Rydberg constant.
The laser: The laser operates on the principle of stimulated emission, by which many photons with identical wavelength and phase are emitted. Laser operation requires a nonequilibrium condition called population inversion, in which more atoms are in a higher-energy state than are in a lower-energy state.
Blackbody radiation: The total radiated intensity (average power radiated per area) from a blackbody surface is proportional to the fourth power of the absolute temperature : (Stefan-Boltzmann law). The quantity is called the Stefan-Boltzmann constant. The wavelength at which a blackbody radiates most strongly is inversely proportional to : (Wien displacement law). The Planck radiation law gives the spectral emittance (intensity per wavelength interval in blackbody radiation): .
The Heisenberg uncertainty principle for particles: The same uncertainty considerations that apply to photons also apply to particles such as electrons. The uncertainty in the energy of a state that is occupied for a time is given by equation .