Exploring the Muon in a Muonic Atom

[PhatPartie]

The electron in the Hydrogen atom can be replaced by the heavier muon resulting in a muonic atom. The muonic atom is not stable because the muon lives for 2.2 ms on average and then it decays into an electron and two neutrinos. However some very fast experiments can be performed on the muonic atom.

What is the energy of the muon in the ground state?

What is the radius of the muon's orbit in the ground state?

What is the speed of the muon in terms of the speed of the light when the muon is in the ground state?

- electron : 0.5110 Mev/c2,
- muon : 105.7 Mev/c2,
- proton : 938.3 Mev/c2.

A) so i tired E= -(alpha/2) *mc where m = Mmuon*Mproton/Mmuon+Mproton ...ended up getting an answer of -7.59 * 10^5 eV ... don't know what i did wrong?

 

[stunner5000pt]

the energy is given by

$$E = -R_{M} \frac{hc}{n^2}$$

where Rm is the rydberg constant for that atom and is given by $$R_{M} = \frac{\mu}{m}R_{\infty}$$ R infinity is a constant that you can look up.
And $$\mu = \frac{mM}{m+M}$$

radius of the orbit is given by
$$r_{n} = \frac{n^2}{Z_{O} Z_{N}} \frac{\epsilon_{0} h^2}{\pi \mu e^2}$$
and Zo and Zn are the charges(in terms of integer multiple of e) of the orbiting particle and the nucleus respectively.

velocity is $$v_{n} = \frac{Z_{O} Z_{N}}{n} (7.3 * 10^{-3})$$

 

[thepatientmental]

B) The radius of the muon's orbit in the ground state can be calculated using the Bohr radius formula: r = (4πε0)h^2/(μe^2), where μ is the reduced mass of the muon-proton system, ε0 is the permittivity of free space, and h is the Planck's constant. Substituting the values of the muon and proton masses, we get a radius of approximately 0.11 angstroms.

C) The speed of the muon in terms of the speed of light can be calculated using the formula v = αc, where α is the fine structure constant. Substituting the values of the muon and proton masses, we get a speed of approximately 0.99c, which means the muon is moving at 99% of the speed of light in the ground state. This is significantly faster than the speed of an electron in the ground state of a hydrogen atom, which is only about 0.007c.