Quantized Energy StatesThe electrons in free atoms can will be found in only certain discrete energy states. These sharp energy states are associated with the orbits or shells of electrons in an atom, e.g., a hydrogen atom. One of the implications of these quantized energy states is that only certain photon energies are allowed when electrons jump down from higher levels to lower levels, producing the hydrogen spectrum.

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Angular Momentum QuantizationIn the Bohr model, the wavelength associated with the electron is given by the DeBroglie relationshipand the standing wave condition that circumference = whole number of wavelengths. These can be combined to get an expression for the angular momentum of the electron in orbit.
Thus L is not only conserved, but constrained to discrete values by the quantum number n. This quantization of angular momentum is a crucial result and can be used in determining the Bohr orbit radii and Bohr energies. 
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Classical Electron Orbit

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Bohr OrbitCombining the energy of the classical electron orbit with the quantization of angular momentum, the Bohr approach yields expressions for the electron orbit radii and energies:Substitution for r gives the Bohr energies and radii:

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Wavelengths for Different States

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Failures of the Bohr ModelWhile the Bohr model was a major step toward understanding the quantum theory of the atom, it is not in fact a correct description of the nature of electron orbits. Some of the shortcomings of the model are: 1. It fails to provide any understanding of why certain spectral lines are brighter than others. There is no mechanism for the calculation of transition probabilities. 2. The Bohr model treats the electron as if it were a miniature planet, with definite radius and momentum. This is in direct violation of the uncertainty principle which dictates that position and momentum cannot be simultaneously determined. The Bohr model gives us a basic conceptual model of electrons orbits and energies. The precise details of spectra and charge distribution must be left to quantum mechanical calculations, as with the Schrodinger equation.

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