Lifetime of Classical Atom

At a time t = 0, the electron orbits a classical hydrogen atom at a radius a₀ equal to the first Bohr radius. Derive an expression for the time it takes for the radius to decrease to zero due to radiation. Assume that the energy loss per revolution is small compared to the total energy of the atom.

SOLUTION

If the energy loss per revolution is small compared to the total energy of the electron in the atom, we can write

(1)

where ω is the acceleration of the electron and P_rad is the total radiated power. Using our assumption, we can approximate the orbit of the electron (which is a spiral) by a circle for each revolution of radius r = r(t). The acceleration is due to the Coulomb force

(2)

On the other hand, using |ε| = |U|/2 (U is the potential energy of a particle moving in a circle in a 1/r² field) we have

(3)

Substituting (2) and (3) into (1) gives

(4)

or

(5)

Integrating (5) yields

So