In Bohr's model of hydrogen the electron moves in a circle of radius at speed

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Origins of quantum theory · Origins of quantum theory

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Matter waves and wave-particle duality

Problem

In Bohr's model of hydrogen the electron moves in a circle of radius

r

at speed

v

, slowly enough that its de Broglie wavelength is

\lambda = h/(m_ev)

. Bohr's orbits have radii

r_n = n^2a_0

with

a_0 = 0.0529

nm, and the electron's kinetic energy in the

n

th orbit is

K_n = 13.6\ \mathrm{eV}/n^2

. ((a)) Show that the condition that a whole number

n

of wavelengths fits round the circumference,

2\pi r = n\lambda

, is the same statement as Bohr's rule

m_evr = n\hbar

. ((b)) Compute the circumference of the

n = 3

orbit, in nm. ((c)) Compute the de Broglie wavelength of an electron whose kinetic energy is

K_3

, in nm, and hence the number of wavelengths that fit round the

n = 3

orbit.