Discussion
The Bohr radius of the hydrogen atom and the Compton wavelength of electron are given by :
Answer: A
The answer is \(a \propto \frac{\hbar^{2}}{me^{2}}\) and \(\lambda_{c} \propto \frac{h}{m_{0}c}\).
The Bohr radius comes from balancing the Coulomb attraction against quantised angular momentum. In Gaussian units
\[ a_{0} = \frac{\hbar^{2}}{me^{2}} \]
which is about 0.53 Å (in SI units \(4\pi\epsilon_{0}\hbar^{2}/me^{2}\)).
The Compton wavelength is the wavelength of a photon whose energy equals the electron's rest energy:
\[ \lambda_{c} = \frac{h}{m_{0}c} \]
which is about \(2.43 \times 10^{-12}\) m.
The other options give the wrong powers: \(\hbar^{2}/m_{0}c\) and \(\hbar/m_{0}c^{2}\) do not have the dimensions of length, and \(\hbar^{2}/mc^{2}\) has no charge in it, although the Bohr radius depends on the electric force.
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