A muon has the same charge as the electron but a mass

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · One-electron atoms and the origins of atomic structure

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The hydrogen atom

Problem

m_\mu = 207\,m_e

. A muonic hydrogen atom is a muon bound to a proton (

m_p = 1836\,m_e

) by the same Coulomb potential energy as ordinary hydrogen. As with positronium, replacing the electron by a particle of mass

m

multiplies the energy scale by

m/m_e

; because

a_0 = 4\pi\epsilon_0\hbar^2/(m e^2)

has

m

in the denominator, the same replacement multiplies the Bohr radius by

m_e/m

. Take

13.6

eV and

a_0=0.0529

nm as the hydrogen energy and length scales, and the proton's own charge radius as

0.88

fm. ((a)) Using the reduced mass

m_r = m_1m_2/(m_1+m_2)

(quoted, as in the rigid-rotor problems), find

m_r

for the muon-proton system, as a multiple of

m_e

, to four significant figures. ((b)) Find the ground-state energy of muonic hydrogen, in keV, and its Bohr radius, in fm. ((c)) Compare the muonic Bohr radius with the proton's charge radius, and separately compare the ordinary (electronic) Bohr radius with the proton's charge radius. Give both ratios. ((d)) Say, in a sentence or two, whether treating the proton as a point charge, an idealisation this chapter makes throughout, is still as good an approximation for muonic hydrogen as it is for ordinary hydrogen.

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