In muonic hydrogen

CHEMISTRY · Multi-part · CBSE · Chemistry · Structure of Atom · Structure of Atom

Problem

In muonic hydrogen, the electron is replaced by a negatively charged muon (

\mu^-

) that has the same charge magnitude as the electron but a mass

m_{\mu}=207\,m_e

. The muon orbits a proton in a circular stationary state described by a principal quantum number

n

. Assume the proton is infinitely massive compared with the muon. Constants: •

e = 1.60\times10^{-19}\,\text{C}

•

\varepsilon_0 = 8.85\times10^{-12}\,\text{C}^2\,\text{N}^{-1}\,\text{m}^{-2}

•

h = 6.63\times10^{-34}\,\text{J\,s}

•

m_e = 9.11\times10^{-31}\,\text{kg}

(a) Derive a general expression for the allowed orbital radius

r_n^{(\mu)}

of the muon in terms of

n

m_{\mu}

, and fundamental constants. Express your result as a multiple of the Bohr radius

a_0

. (b) Calculate the ground-state (

n=1

) radius of muonic hydrogen and compare it quantitatively with the ground-state radius of ordinary hydrogen. (c) Determine the ground-state energy

E_{1}^{(\mu)}

of muonic hydrogen in electron-volts (eV). (d) Explain qualitatively how the smaller orbital radius affects the wavelengths of spectral lines in muonic hydrogen compared with ordinary hydrogen.

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