Quantum mech help - radii of some atoms

[yxgao]

What is the ratio of the radii of muonic hydrogen to hydrogen? Muonic hydrogen consists of a proton and $$\mu^{-}$$. How do you calculate this??

 

[dextercioby]

What is the ratio of the radii of muonic hydrogen to hydrogen? Muonic hydrogen consists of a proton and $$\mu^{-}$$. How do you calculate this??

The concept of Radius of an atom is itself entirely classical,so when reading the text of the problem u should have pictured Bohr's semiclassical model for finite mass nucleus.
U'll be neding this formula
$$R_{n}=\frac{1}{Z}\frac{4\pi\epsilon_{0}\hbar^{2}}{\mu e^{2}} n^{2}$$
,where
$$\frac{1}{\mu}=\frac{1}{m_{nucleus}}+\frac{1}{m_{electron/muon}}$$

Daniel.

 

[wais]

The ratio of the radii of muonic hydrogen to hydrogen can be calculated using the Bohr radius formula, which is given by r = n^2 * h^2 / (4 * pi * m * k * e^2), where n is the principal quantum number, h is Planck's constant, m is the reduced mass of the system, k is the Coulomb constant, and e is the elementary charge.

For muonic hydrogen, the reduced mass is given by m = m_p * m_\mu / (m_p + m_\mu), where m_p is the mass of a proton and m_\mu is the mass of a muon.

Using this formula, we can calculate the radii of muonic hydrogen and hydrogen, and then take the ratio of these radii to find the desired ratio. It is important to note that the principal quantum number, n, for muonic hydrogen will be different from that of regular hydrogen due to the difference in mass of the particles.

Overall, the ratio of the radii of muonic hydrogen to hydrogen will be less than 1, as the reduced mass of the system is larger for muonic hydrogen, resulting in a smaller Bohr radius. This difference in radii can also be attributed to the muon having a larger mass than an electron, leading to a stronger attraction between the particles and a smaller orbit for the muon.