The size of a hydrogen atom in its ground state is judged to depend on the reduced…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Quantities, units and the practice of measurement · Quantities, units and the practice of measurement

Measurement and Method, Ch What dimensions decide before you measure anything

Problem

The size of a hydrogen atom in its ground state is judged to depend on the reduced Planck constant

\hbar

(unit

\mathrm{J\,s}

), the electron mass

m_{\rm e}

(unit

\mathrm{kg}

), and the combination

k = e^2/(4\pi\epsilon_0)

, where

e

is the elementary charge and

\epsilon_0

the permittivity of free space;

k

appears because the Coulomb potential energy of the electron is

-k/r

at separation

r

, so

k

has the dimension of energy

\times

length. ((a)) Write the Bohr radius as

a_0 = C\hbar^{\alpha}m_{\rm e}^{\beta}k^{\gamma}

with

C

dimensionless, and use dimensional homogeneity to find

\alpha

\beta

and

\gamma

. ((b)) Taking the value

C = 1

(supplied by the full theory, not by this method), evaluate

a_0

numerically using

\hbar = 1.055\times10^{-34}\ \mathrm{J\,s}

m_{\rm e} = 9.109\times10^{-31}\ \mathrm{kg}

e = 1.602\times10^{-19}\ \mathrm{C}

and

\epsilon_0 = 8.854\times10^{-12}\ \mathrm{F\,m^{-1}}

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