Question

Calculate the separation between the particles of a system in the ground state, the corresponding binding energy, and the wavelength of the first line of Lyman series,if such a system is
(a) a mesonic hydrogen atom whose nucleus is a proton (in a mesonic atom an electron is replaced by a meson whose charge is the same and mass is `207` si that of an electron),
(b) a positronium consisting of an electron and positron revolving around their common centre of masses.

Answer 1

In a mesonic system, the reduced mass of the system is related to the mass of the meson `(m_(mu))` and proton `(m_(p))` by
`mu=(m_(mu)m_(p))/(m_(mu)+M_(p))= 186.04 m_(e )`
Then,
separation between the particles in the ground state `=( ħ^(2))/(mue^(2))`
`=(1)/(186) ( ħ^(2))/(m e^(2))`
`E_(b)=(meason) =(mu e^(4))/(2 ħ^(2))= 186xx13.65eV`
`=2.54 keV`
`lambda_(1)=(8 pi ħc)/(3E_(b)(meason))=(lambda_(1)(Hydr og e n))/(186)= 0.65nm`
(on using `lambda_(1)(Hydrogr en)=121nm)`
(b) In the postitronium
`mu=(m_(e )^(2))/(2m_(e ))=(m_(e ))/(2)`
The sepration between the particles is the ground state
`=2(ħ^(2))/(m_(e )e^(2))= 105.8p m`
`E_(b)(positronium)=(m_(e))/(2).(e^(4))/(2ħ^(2))=(1)/(2)E_(b)(H)=6.8eV`
`lambda_(1)(postironium)=(2lambda_(1)(Hydrog en)=0.243nm`