I need some help with these questions on de Broglie hypothesis: (See attachment for better formula symbols)
1.a. Show how the de Broglie hypothesis for the wavelength of an electron can lead to an explanation of the condition for quantization of orbital angular momentum in the Bohr model for hydrogen atom.
b. Show that the frequency of the radiation emitted by an electron in the hydrogen atom when it makes transition from the n+1 state to the n state given by: f(n, n+1) = mk^2e^4 (2n+1) / 4pih^3 n^2(n+1)^2 where k is the Coulomb constant, and m and e are the mass and charge of the electron, respectively.
c. Show that for a very large n, the above frequency becomes equal to the orbital frequency of the electron. What principle does this result illustrate?
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The stability of the Bohr atom electron orbits could be explained in terms of De Broglie's
wave like character of matter. The idea is that the orbits of the electrons about the
hydrogen atom nucleus are an integer number of wavelength in a similar condition to the
formation of a standing wave in a string. In this case the circumference of the orbits is
equal to 2 П r . The de broglie wavelength is λ = h / mv . Since to form a standing wave
we need integer number of wavelengths in the circumference , therefore
n λ = 2 П r ........................................................... ( 1 )
Combining these two ...
This in-depth solution explains how the de Broglie hypothesis can be applied in the Bohr model of the hydrogen atom, frequency of radiation of the electron when it makes transition of state, and also how a large n causes the above frequency to equal orbital frequency.