# Neumann's Principle - applicable to the wavefunction?

Neumann's Principle states:

If a crystal is invariant with respect to certain symmetry elements, any of its physical properties must also be invariant with respect to the same symmetry elements.

So, does this apply to the wavefunction of an electron in a crystal? Or, stated another way, if the crystal is an eigenlattice of some symmetry operator, is the wavefunction necessarily an eigenfunction of the same operator?

ZapperZ
Staff Emeritus
Neumann's Principle states:

If a crystal is invariant with respect to certain symmetry elements, any of its physical properties must also be invariant with respect to the same symmetry elements.

So, does this apply to the wavefunction of an electron in a crystal? Or, stated another way, if the crystal is an eigenlattice of some symmetry operator, is the wavefunction necessarily an eigenfunction of the same operator?

It certainly does! For example, the translational symmetry of a crystal at each lattice points results in the Bloch wave function. In fact, I would even say that the concept of symmetry and broken symmetry is one of the most important concept in condensed matter (see the origin of Higgs mechanism).

Zz.

It certainly does! For example, the translational symmetry of a crystal at each lattice points results in the Bloch wave function. In fact, I would even say that the concept of symmetry and broken symmetry is one of the most important concept in condensed matter (see the origin of Higgs mechanism).

Zz.

Wow thanks for the answer. That's a very powerful concept, I began thinking about this when someone showed me how you can determine the form of the electric susceptibility tensor of a crystalline material just by considering the crystal symmetry, I was very impressed.

The wave functions of crystals are usually given in reciprocal space, within the first Brillouin zone.

For each point within the BZ, the wave function at that point must have the symmetry of the reciprocal space point. For high symmetry points, e.g. on the BZ boundary and on high-symmetry axes one can derive a lot of properties from symmetry alone.

Also, one usually has to calculate only a small fraction of the BZ, as the rest can be derived by applying symmetries. For a high-symmetry crystal like Si with 48(!) symmetries the savings are considerable.