A single electron of mass and charge orbits a fixed nucleus of charge

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Origins of quantum theory

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Line spectra, the Bohr atom and the old quantum theory

Problem

A single electron of mass

m_e

and charge

-e

orbits a fixed nucleus of charge

+Ze

, the only force being the Coulomb attraction

Ze^2/4\pi\epsilon_0r^2

. Bohr's rule is

m_evr = n\hbar

, and light is emitted only in a jump with

h\nu = E_{n_2} - E_{n_1}

. Write

|E_1^{\mathrm H}| = 13.6

eV for hydrogen's ground-state binding energy, and take

h = 6.626\times10^{-34}

J s,

1\ \mathrm{eV} = 1.602\times10^{-19}

J. Use the binomial expansion

(1 - x)^{-2} \approx 1 + 2x

for small

x

where needed. ((a)) Show that

r_n = n^2a_0/Z

and

E_n = -Z^2|E_1^{\mathrm H}|/n^2

. ((b)) Show that for large

n

the frequency of the

n \to n-1

line tends to

2Z^2|E_1^{\mathrm H}|/hn^3

, and that this is also the classical orbital frequency

v_n/2\pi r_n

of the

n

th orbit. ((c)) For He

^+

(

Z = 2

) with

n = 200

, compute the exact Bohr line frequency of the

200 \to 199

jump and the classical orbital frequency of the

n = 200

orbit, both in GHz.