Are Eigenfunctions of a Hamiltonian on a lattice always Bloch functions?

In summary, it is well known in Physics that eigenstates possess the same symmetries as the Hamiltonian. This is easily proven by showing that the eigenspaces are closed under the operations of the symmetry group. To further explore this concept, one may consider decomposing the eigenspace into eigenstates or using Schur's lemma. One question that arises is whether every Eigenfunction of a Hamiltonian is invariant under the operation of the Hamiltonian's symmetry groups, up to a scale factor of unity magnitude. This question can be answered using spherical harmonics and it is determined that only the eigenspaces are invariant. It can also be argued that on a lattice, the eigenfunctions must be Bloch functions.
  • #1
0xDEADBEEF
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It is common knowledge in Physics that eigenstates share the symmetries of the Hamiltonian.

And it is trivial to show that this is true for the eigenspaces. Let g be an element of a symmetry group of Hamiltonian H, [tex]M_g[/tex] its representation, [tex]\left| \phi \right>[/tex] an eigenvector and [tex]\lambda[/tex] the corresponding eigenvalue. If g is an element from a symmetry group of H it is given that:

[tex]M_gHM_{g^{-1}}\left|\phi\right>=\lambda\left|\phi\right>[/tex]
Thus
[tex]HM_{g^{-1}}\left|\phi\right> = \lambda M_{g^{-1}}\left|\phi\right>[/tex]

So we see that the eigenspace for [tex]\lambda[/tex] is closed under the operations of [tex]M_g[/tex]
Is there some theorem, that I can decompose this eigenspace into eigenstates of M or how do I proceed from here? This is important for Bloch functions and crystallography. I read something about Schur's lemma being involved.

So to phrase a proper question:
Is every Eigenfunction of a Hamiltonian,invariant up to a scale factor of unity magnitude under the operation of the Hamiltonian's symmetry groups? And if this is so how do I show this?
 
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  • #2
Ok I think I have answered my question using spherical harmonics. It is really only the eigenspaces that are invariant. But then I don't know how to argue that on a lattice the eigenfunctions must be Bloch functions.
 

1. What is meant by "symmetry of eigenstates"?

The symmetry of eigenstates refers to the pattern or structure of the wavefunction of a quantum system that describes its energy eigenstates. These eigenstates are solutions to the Schrödinger equation and their symmetry is determined by the symmetry of the potential energy function that describes the system.

2. How is symmetry related to the properties of eigenstates?

The symmetry of eigenstates is directly related to the properties of the system. In quantum mechanics, symmetry is a fundamental concept that determines the allowed energy levels and transitions of a system. The properties of eigenstates, such as their energy and angular momentum, are determined by the symmetry of the system.

3. What are the different types of symmetries that can be found in eigenstates?

There are three main types of symmetries that can be found in eigenstates: translational symmetry, rotational symmetry, and reflection symmetry. These symmetries can be further broken down into subgroups, such as discrete or continuous symmetries, depending on the specific system.

4. How does symmetry affect the degeneracy of energy eigenstates?

Symmetry plays a crucial role in determining the degeneracy of energy eigenstates. In general, systems with higher symmetry have a higher degeneracy, meaning that there are more energy eigenstates with the same energy level. This is because symmetrical systems have more constraints, leading to more possible solutions to the Schrödinger equation.

5. Can symmetries be broken in quantum systems?

Yes, symmetries can be broken in quantum systems. This can happen due to external influences, such as the presence of a magnetic field, or due to the interaction between particles. In these cases, the symmetries of the system may no longer be preserved, leading to a different set of eigenstates and energy levels.

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