# 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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Contributed by: S. M. Blinder (March 2019)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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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