Past exam of the quantum condensed matter course of the University of Cambridge / 2022 / Section B / 4 / b

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- Solution b

Solution

Substitute harmonic amplitudes with the specified

e^{- iω t}

convention:

m (ω_{0}^{2} - ω^{2} - iγω) u_{0} = q E_{0} .

One electron contributes a dipole moment

p = q u_{0}

, so the electric polarization of a density

n

is

P_{0} = n q u_{0} = \frac{n q ^{2}}{m ( ω _{0}^{2} - ω ^{2} - iγω )} E_{0} .

Using

D = ϵ_{0} E + P = ϵ_{0} ϵ_{r} E

gives

ϵ_{r} (ω) = 1 + \frac{ω _{p}^{2}}{ω _{0}^{2} - ω ^{2} - iγω}, ω_{p}^{2} = \frac{n q ^{2}}{m ϵ _{0}} .

Thus the paper's dimensionless

ϵ_{ω}

is the relative permittivity, rather than the absolute permittivity

ϵ_{0} ϵ_{r}

. The electron's negative charge disappears from the electric susceptibility because both the force and dipole moment contain

q

.

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