Question

A tritium atom (³H) can undergo spontaneous radioactive decay into a helium-3 ion (³He⁺) by emission of a beta particle. The departure of the electron is so fast that to the orbital electron the process appears as simply an instantaneous change in the nuclear charge from Z = 1 to Z = 2. Calculate the probability that the He ion will be left in its ground state.

Answer 1

The wave function of the ground state of He⁺ is

where a is the Bohr radius. Let the wave function of ³H be \(\phi\)(r).

As the process of \(\beta\) decay takes place very fast, during the time period in which the ³H becomes ³He⁺ the wave function does not have time to change. Hence the probability that the ³He⁺ is in the ground state is

Initially, the ³H is in the ground state so that