For an electron (mass , charge ) bound by a harmonic potential and acted upon by a constant external electric field , the Schrödinger equation can be written as
An exact solution can be obtained by completing the square in the potential energy :
Introducing the new variable , the Schrödinger equation can be written as
making use of the known solution of the standard harmonic-oscillator problem, expressed in terms of . The perturbed energies are shifted downward by a constant term:
The graphic shows the potential energy and energy levels for the unperturbed (in black) and perturbed (in red) oscillator, for selected values of and . For simplicity, atomic units, , are used.
If the electric field is turned on during a time interval that is short compared to the oscillation period , the sudden approximation in perturbation theory can be applied . Accordingly, the transition probability from state to a state is given by . These results can be seen by selecting "show transition probabilities" and the initial state .