Summarizing the Wave Nature of Matter Causes Quantization

The Wave Nature of Matter Causes Quantization Summary

• Quantization of orbital energy is caused by the wave nature of matter. Allowed orbits in atoms occur for constructive interference of electrons in the orbit, requiring an integral number of wavelengths to fit in an orbit’s circumference; that is,

  \({\mathrm{n\lambda }}_{n}={2\mathrm{\pi r}}_{n}(n=1, 2, 3 ...),\)

  where \({\lambda }_{n}\) is the electron’s de Broglie wavelength.
• Owing to the wave nature of electrons and the Heisenberg uncertainty principle, there are no well-defined orbits; rather, there are clouds of probability.
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by

  \(\Delta E=\text{hf}={E}_{\text{i}}-{E}_{\text{f}},\)

  where \(\Delta E\) is the change in energy between the initial and final orbits and \(\text{hf}\) is the energy of an absorbed or emitted photon.
• It is useful to plot orbit energies on a vertical graph called an energy-level diagram.
• The allowed orbits are circular, Bohr proposed, and must have quantized orbital angular momentum given by

  \(L={m}_{e}{\text{vr}}_{n}=n\cfrac{h}{2\pi }(n=1, 2, 3 ...),\)

  where \(L\) is the angular momentum, \({r}_{n}\) is the radius of orbit \({n}^{}\), and \(h\) is Planck’s constant.