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Appendix B Solutions

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Table of Contents

Basic

Medium

Basic

B-1. SI Calculation of the Electron Compton Wavelength

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\(\lambda_C = \frac{\hbar}{m_e c}\)

\(= \frac{1.055 \times 10^{-34}}{9.109 \times 10^{-31} \times 2.998 \times 10^8}\)

\(= \frac{1.055 \times 10^{-34}}{2.731 \times 10^{-22}}\)

\(= 3.86 \times 10^{-13} \;\text{m} \approx 0.386 \;\text{pm}\)

This is far smaller than the size of an atom (\(\sim 10^{-10}\) m) and larger than the size of a nucleus (\(\sim 10^{-15}\) m). The Compton wavelength represents "the scale at which quantum mechanical effects become important."

B-2. Planck Mass in GeV

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\(m_P = \sqrt{\frac{\hbar c}{G}} = 2.176 \times 10^{-8} \;\text{kg}\)

Converting to GeV:

\(m_P c^2 = 2.176 \times 10^{-8} \times (2.998 \times 10^8)^2 = 1.956 \times 10^9 \;\text{J}\)

\(= \frac{1.956 \times 10^9}{1.602 \times 10^{-10}} \;\text{GeV} = 1.221 \times 10^{19} \;\text{GeV}\)

Ratio to the proton mass \(m_p \approx 0.938\) GeV:

\(\frac{m_P}{m_p} = \frac{1.221 \times 10^{19}}{0.938} \approx 1.3 \times 10^{19}\)

The Planck mass is approximately \(10^{19}\) times the proton mass.

Medium

M-1. Dimensions of \(G\) in Natural Units

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Dimensions of \(G\) in SI units:

\([G] = \frac{[\text{length}]^3}{[\text{mass}][\text{time}]^2} = \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\)

In natural units (\(\hbar = c = 1\)): - From \(c = 1\): \([\text{length}] = [\text{time}]\) - From \(\hbar = 1\): \([\text{energy}] \cdot [\text{time}] = 1\), i.e., \([\text{time}] = [\text{energy}]^{-1}\) - From \(E = mc^2\): \([\text{mass}] = [\text{energy}]\)

Therefore \([\text{length}] = [\text{time}] = [\text{energy}]^{-1}\), \([\text{mass}] = [\text{energy}]\).

\([G] = \frac{[\text{E}]^{-3}}{[\text{E}][\text{E}]^{-2}} = \frac{[\text{E}]^{-3}}{[\text{E}]^{-1}} = [\text{E}]^{-2}\)

Indeed \([G] = [\text{energy}]^{-2}\).

M-2. Fine Structure Constant in Natural Units

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Definition in SI units:

\(\alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c}\)

In natural units where \(\hbar = c = 1\), and further adopting the Gaussian unit convention \(4\pi\varepsilon_0 = 1\):

\(\alpha = \frac{e^2}{4\pi}\)

Since \(\alpha \approx 1/137\) is a dimensionless quantity, its value is the same in any unit system. In natural units, \(e^2 = 4\pi\alpha \approx 4\pi/137 \approx 0.0917\).