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Charged Harmonic Oscillator in Electric Field

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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

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.

An exact solution can be obtained by completing the square in the potential energy [1]:

.

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 [2]. 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 .

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Contributed by: S. M. Blinder (March 2019)
Open content licensed under CC BY-NC-SA

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Details

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.

References

[1] 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.

[2] 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.

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