Charged Harmonic Oscillator in Electric Field
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[more]
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 .[less]
The unperturbed harmonic oscillator has the eigenfunctions (in atomic units):
where is a Hermite polynomial. The perturbed oscillator has an analogous form, with a shifted argument:
The transition probability according to the sudden approximation is given by , where.
 C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, Vol. I (S. R. Hemley, N. Ostrowsky and D. Ostrowsky, trans.), New York: Wiley, 1977 pp. 552–554.
 L. D. Landau and E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory, 2nd ed. (J. B. Sykes and J. S. Bell, trans.), New York: Pergamon Press, 1965 pp. 142–143.