[물리] Positronium [Berkeley]

Positronium is a hydrogen atom but wit a positron as nucleus instead of a proton. In the nonrelativistic limit, the energy levels and wave functions are the same as for hydrogen, except for scale.

(a) From your knowledge of the hydrogen atom, write down the normalized wave function for the 1s ground state of positronium. Use spherical coordinates and the hydrogenic Bohr radous a_0 as a scale parameter

sol)

By analogy with the hydrogen atom the normalized wave function for the 1s ground state of positronium is

, m being the electron rest mass. Note that the factor in 2 in front of a_0 is to account for the factor that the reduced mass is μ = (1/2)m.

(b) Evaluate the root-mean-square radius for the 1s state in units of a_0. Is this an estimate of the physical diameter or the radius of positronium?

sol)

The mean-square radius for the 1s state is

and the rms radius is 

(c) In the s states of positronium, there is a contact hyperfine interaction

where μ_e and μ_p are the electron and positron magnetic moments

For electrons and positrons, |g-factor| = 2. Using first order perturbation theory, compute the energy difference between the singlet and triplet ground states.

Determine which state lies lowest. Express the energy splitting in GHz.

sol)

We know that n, l, m, S, Sz are good quantum numbers. Thus,

where we have used S = s_e + s_p, and so

For singlet state, S = 0, S_z = 0,

For triplet state, S = 1, S_z = 0, ±1,

Thus the singlet ground state has the lowest energy and the energy splitting of the ground state is