A nonrelativistic electron of charge —e and mass m bound in an attractive Coulomb potential (–Ze²/r) moves in a circular orbit in the absence of radiation reaction.

(a) Show that both the energy and angular-momentum equations (16.13) and (16.16) lead to the solution for the slowly changing orbit radius,

r³(t) = r³₀ – 9Z(cτ)³ t/τ

where r0 is the value of r(t) at t = 0.

(b) For circular orbits in a Bohr atom the orbit radius and the principal quantum number n are related by r = n²a₀/Z. If the transition probability for transitions from n à (n – 1) is defined as –dn/dt, show that the result of part a agrees with that found in Problem 14.21.

(c) From part a calculate the numerical value of the times taken for a mu meson of mass m = 207m_e to fall from a circular orbit with principal quantum number n₂ = 10 to one with n₂ = 4, and n₂ = 1. These are reasonable estimates of the time taken for a mu meson to cascade down to its lowest orbit after capture by an isolated atom.