Very Small Physics 2

Hydrogen is the simplest atom, with only one proton and one electron interacting with a Coulomb potential of \[U = -\frac{e^2}{4 \pi \varepsilon_0 r}.\] If the electron were a classical charged particle, its orbit around the nucleus would decay until it came to rest at the nucleus. Since the electron is a quantum object, this doesn't happen. One way to estimate the minimum radius allowed for a quantum electron is by saturating the Heisenberg bound, which puts a lower limit on the radius of the electron's orbit \(r\) and its momentum \(p\): \[r p \approx \Delta x \Delta p \approx \hbar.\] Treat the hydrogen atom as a two-body problem where the mass of the proton \(m_p\) is much greater than that of the electron \(m_e.\)

Estimate the minimum radius of the electron orbit \(r_\text{min}.\)

Details: \(e\) is the charge of an electron, \(\varepsilon_0\) is the permittivity of free space, and \(\hbar\) is the reduced Planck's constant.