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.


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