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
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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