Bohr's hydrogen levels

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · 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

Bohr's hydrogen levels, derived from

m_evr = n\hbar

and the Coulomb force, are

E_n = -|E_1|/n^2

with

|E_1| = 13.6

eV, and light is emitted only in a jump with

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

. Use

hc = 1240

eV nm and the binomial expansion

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

for small

x

. ((a)) Show that for large

n

the frequency of the

n \to n-1

line is

\nu \approx 2|E_1|/hn^3

, and hence that its wavelength is

\lambda \approx hcn^3/2|E_1|

. ((b)) Use that approximation to find the wavelength of the

30 \to 29

line of hydrogen, in mm. ((c)) Find the same wavelength exactly from

\Delta E = E_{30} - E_{29}

, in mm, and state the percentage by which the approximation is too large.