Dipole moment of an isolated quantum system

[Sheldon Cooper]

How to prove the dipole moment of an isolated quantum system in isotropic space is identically equal to zero, unless there exists an accidental degeneracy.

Thanks in advance

 

[Cryo]

Do you mean electric dipole or any dipole (I know of three)?

Assuming you mean electric dipole, is your statement true? Is a piece of ferroelectic in vacuum an isolated quantum system? Afterall, it could have electric dipole moment?

For simple systems, like particles I think the argument usually boils down to ground-state being symmetric, and therefore unaffected by space inversion, whereas electric dipole being a polar vector and therefore affected by space inversion, thus the expectation value for the electric dipole has to be equal to its negative self -> it's zero.

 

[king vitamin]

Assuming you mean electric dipole, is your statement true? Is a piece of ferroelectic in vacuum an isolated quantum system? Afterall, it could have electric dipole moment?

As an aside, the example of an electric dipole moment in was precisely what Philip Anderson used in his famous paper "More Is Different" to demonstrate the notion of the emergence of new laws of physics and symmetry breaking in complex systems. He had learned in a nuclear physics course in graduate school that no stationary state in a parity invariant theory has a net electric dipole moment, which confused him because there seem to be numerous examples to the contrary away from nuclear physics. His thinking on this led him to important insights about how symmetry can be spontaneously broken in quantum systems (he was one of the pioneers of understanding symmetry breaking in physics).

The resolution is that the ferroelectric is not actually in an eigenstate. The true eigenstates are the two quantum superpositions of the states with equal/opposite dipole moments (and therefore zero dipole moment)*, and there is some time period (probably much longer than the age of the universe) for the rotation of the macroscopic dipole moment. This classic paper by Anderson on symmetry breaking in antiferromagnets was the first detailed description of this phenomenon.

* Here I'm assuming that we can ignore parity-violating effects in particle physics for the purposes of describing stable matter.