a) Using Heisenberg’s uncertainty principle, estimate the energy and radius of the ground state of a hydrogen-like atom made of a proton and a muon. Assume that the 

proton and the muon are bound by the Coulomb potentia  The mass of the muon is 

To estimate the energy and radius of the ground state of a hydrogen-like atom made of a proton and a muon using the Heisenberg uncertainty principle, we need to consider the uncertainty relation between position and momentum, and then relate it to the energy of the system.

The uncertainty principle states:

\[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\]

For the ground state of a hydrogen-like atom, the uncertainty in position (\(\Delta x\)) can be considered approximately as the radius of the orbit (\(r\)), and the uncertainty in momentum (\(\Delta p\)) can be considered approximately as the momentum of the electron (\(p\)).

The momentum of the muon (\(p\)) can be related to its energy (\(E\)) using the relativistic energy-momentum relation:

\[E^2 = (pc)^2 + (m_0 c^2)^2\]

where \(m_0\) is the rest mass of the muon and \(c\) is the speed of light.

Given that the mass of the muon (\(m_0\)) is \(106 \, \text{MeV}/c^2\), we can find the momentum (\(p\)).

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