# Particles Behaving as Waves

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: $\lambda=\frac hp=\frac h{mv}$, $E=hf$.  A non-relativistic electron accelerated from rest through a potential difference $V_{ba}$ has a wavelength $\lambda=\frac hp=\frac h{\sqrt{2meV_{ba}}}$.  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 $E_i$ to a lower level $E_f$, it emits a photon of energy $E_i-E_f$: $hf=\frac{hc}{\lambda}=E_i-E_f$.  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 $h/2\pi$: $L_n=mv_nr_n=n\frac{h}{2\pi}$, $(n=1,2,3,\ldots)$.  The integer multiplier $n$ is called the principal quantum number for the level.  The orbital radii are proportional to $n^2$: $r_n=\epsilon_0\frac{n^2h^2}{\pi me^2}=n^2a_0$, $v_n=\frac{1}{\epsilon_0}\frac{e^2}{2nh}$.  The energy levels of the hydrogen atoms are given by $E_n=-\frac{hcR}{n^2}=-\frac{13.60\,\mathrm{eV}}{n^2}$, $(n=1,2,3,\ldots)$, where $R$ 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 $T$: $I=\sigma T^4$ (Stefan-Boltzmann law).  The quantity $\sigma=5.67\times 10^{-8}\,\mathrm{W/m^2\cdot K^4}$ is called the Stefan-Boltzmann constant.  The wavelength $\lambda_m$ at which a blackbody radiates most strongly is inversely proportional to $T$: $\lambda_mT=2.90\times 10^{-3}\,\mathrm{m\cdot K}$ (Wien displacement law).  The Planck radiation law gives the spectral emittance $I(\lambda)$ (intensity per wavelength interval in blackbody radiation): $I(\lambda)=\frac{2\pi hc^2}{\lambda^5(e^{hc/\lambda kT}-1)}$.

The Heisenberg uncertainty principle for particles:  The same uncertainty considerations that apply to photons also apply to particles such as electrons.  The uncertainty $\Delta E$ in the energy of a state that is occupied for a time $\Delta t$ is given by equation $\Delta t\Delta E\geq\hbar/2$.