# Polarizability of Hydrogen Atoms

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An electric field distorts the charge distribution of an atom. For states, the energy is decreased by to second order in the field strength. The parameter is known as the (electric) polarizability. We consider the and states of the hydrogen atom, making use of perturbation theory. The graphic shows a contour plot of the wavefunction as the electric field is increased. The wavefunction is positive in the blue region and negative in the yellow region. A similar distortion of the initially spherical charge density occurs in the formation of chemical bonds.

Contributed by: S. M. Blinder (September 2019)

Open content licensed under CC BY-NC-SA

## Details

For the ground state,

and ,

in atomic units. The unperturbed Hamiltonian is

,

while the perturbation due to an electric field (in the direction) is

.

The first-order perturbation equation for the ground state is given by

The first-order energy . We obtain thereby an inhomogeneous differential equation

.

Writing , the differential equation reduces to

.

The solution that is finite for all is

.

Thus the ground state eigenfunction to first order is given by

.

The second-order energy is obtained from

,

giving a polarizability .

An analogous treatment for the state results in the following approximation:

,

with a polarizability .

Reference

[1] L. I. Schiff, *Quantum Mechanics*, 3rd ed., New York: McGraw-Hill, 1968 pp. 263–266. The approach is attributed to M. Kotani, *Quantum Mechanics*, Vol. I, Tokyo: Yuwanami Book Co., 1951 p. 127.

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