3.I 9D9 \mathrm{D} \quad

Quantum Mechanics | Part IB, 2004

Write down the expressions for the classical energy and angular momentum for an electron in a hydrogen atom. In the Bohr model the angular momentum LL is quantised so that

L=n,L=n \hbar,

for integer nn. Assuming circular orbits, show that the radius of the nn 'th orbit is

rn=n2ar_{n}=n^{2} a

and determine aa. Show that the corresponding energy is then

En=e28πϵ0rnE_{n}=-\frac{e^{2}}{8 \pi \epsilon_{0} r_{n}}

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