Consider an electron in hydrogen atom that is initially at ground state. Uniform electric field along z-axis is applied to this hydrogen atom at t=0.
In polar coordinate,
Here, the constants are and. Now determine the probability of finding the particle in.
Treat the applied electric field as time dependent perturbation.
So that the time independent part of the perturbation is
And the time dependent part is
The first order correction to the energy for time independent perturbation is
Here, the wave functions and represent the first excited state and the ground state of the hydrogen atom and are expressed as:
Substitute the value of functions in perturbation equation (3) and integrate over all positive time.
The probability of the finding the electron in first excited state:
Therefore the probability of finding the electron in the first excited state is