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I've read that there are some restrictions on the value of a possible intrinsic electric dipole possessed by the electron, but isn't the dipole value dependent on the electron's wavefunction? Assuming you can associate a dipole moment to every electronic state, shouldn't the dipole moment roughly be $e<\bar{r}>$?

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Nope. The electron dipole moment is defined as its intrinsic property – a property that may be imagine as a consequence of the particle's internal structure or, more generally, "something that happens inside it". So you may imagine that the relevant wave function for the "center of mass" is $\delta^{(3)}(\vec r)$. It's not literal because one can't localize a particle infinitely accurately but this idea conveys the point that the electron dipole moment is the part of the dipole effects that is independent of its wave function.

You may therefore imagine that the nonzero electric dipole moment means that the "center of mass" and the "center of charge" are located at different points.

More operationally, the electron dipole moment manifests itself as the extra term in the energy $$ U = \vec d_e \cdot \vec E $$ where $\vec d_e$ is the dipole moment vector that is proportional to the spin. So note that the electron with the very same wave function for the center-of-mass will have energy that depends on the spin if there's electric field around.

This inner product of the spin and the electric field is "surprising" because one of them is a vector and the other one is a pseudovector. And indeed, one actually needs a CP violation for this term to be nonzero.

But it is nonzero.

The Standard Model predicts a tiny value, about $10^{-38}$ The unit is the product of the elementary (electron) electric charge and a centimeter. This tiny value arises from virtual effects of quarks that see CP-violations via the CKM matrix's complex phase. New physics may produce new sources of CP-violation so the dipole moment may be much larger. However, experimentally, it's smaller than $10^{-28}$, still tiny. Note that it corresponds to the separation of $+e$ and $-e$ by $10^{-30}$ meters, a near-Planckian tiny distance.

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