Electric Dipole Moment

Definition of Electric Dipole Moment

The Energy of a system of charges is defined as:

ε=∫ρΦ dV
where ρ is the charge density and Φ the potential both functions of coordinates.Choosing a particular origin and expanding the potential in a Taylor series about the origin:

ε=∫ρ{Φ(r₀)+x_i&part_i&Phi(r₀)+½ x_ix_j&part_i&part_j&Phi(r₀)+...}

ε=∫ρΦ(r₀)+d_iE_i+d_ijE_iE_j+...

where d_i is the Electric Dipole Moment and d_ij is the Induced Dipole Moment and E the electric field.

For discrete charges the dipole moment is

d=&Sigma_ir_ie_i

where i denotes the sum over the charges.

EDM for a system
Let P is the Parity operator.If P is conserved then the energy eigenstates are also
parity eigenstates.Under a parity transformation

&lang m|P⁺PdP⁺P|m &rang =-&lang m|d|m &rang
.
So the dipole moment must vanish.A non-vanishing dipole moment signals violation of parity.


Let us consider non-degenerate energy eigenstates characterised by only by its angular momentum operator J.

Under the action of time reversal,

T d T^-1 =d

T J T^-1 =-J

implying the EDM has to vanish.

So, the presense of an electric dipole moment would violate both time reversal and parity.

However, a system possessing non-degenerate eigenstates which are parity eigenstates as well can have a non vanishing dipole moment.This is because the energy shift is quadratic in &Delta² where &Delta is &lang+|zE_z|- &rang and therefore quadratic in E.So it is an induced dipole moment and doesnt imply the violation of time reversal.


When the system is in a degenerate state ( for example the ground state of a water molecule), the energy shift can be linear in E, but then it does not imply violation of time reversal invariance.This is usually called the permanent dipole moment.