Symmetry and Degeneracy in Quantum Systems

1. Symmetry, Commutation, and Degeneracy

The bridge between a system's symmetry and its energy spectrum is built upon a simple but profound commutation relation. A quantum system's Hamiltonian $H$ is said to possess a symmetry described by a group $G$ if it is invariant under all transformations of the group. As we saw last lesson, this invariance of an operator means it must commute with the unitary operators $U(g)$ that represent the group elements $g\in G$.

This commutation has an immediate and crucial consequence. Let $|\psi ⟩$ be an eigenstate of the Hamiltonian with energy $E$, so that $H|\psi ⟩=E|\psi ⟩$. Now, consider a new state $|{\psi }^{\prime }⟩=U(g)|\psi ⟩$, which is the original state transformed by a symmetry operation. Let's see what happens when the Hamiltonian acts on this new state:

Because $H$ and $U(g)$ commute, we can swap their order:

This result is fundamental: the transformed state $|{\psi }^{\prime }⟩$ is also an eigenstate of the Hamiltonian with the exact same energy $E$.

If $|{\psi }^{\prime }⟩$ is linearly independent of $|\psi ⟩$, then we have found a distinct state with the same energy, which is the definition of degeneracy. This implies that the entire set of degenerate eigenstates corresponding to an energy level $E$, known as the eigenspace $V_E$, must be closed under the action of the symmetry group. Any state in the eigenspace, when acted upon by a symmetry operator, yields another state within the same eigenspace.

This means the eigenspace $V_E$ itself carries a representation of the symmetry group $G$.

2. Irreducible Representations and Essential Degeneracy

Now, we can connect this physical observation to the formal machinery of representation theory you'll be studying in this course. While an eigenspace $V_E$ carries a representation of the symmetry group $G$, this representation can be either reducible or irreducible.

However, a key principle of quantum mechanics is that unless there's a special reason (which we'll call an "accidental" degeneracy), a symmetry will not leave degeneracies unexplained. The representation carried by an eigenspace is, in the generic case, irreducible. If it were reducible, one could generally introduce a small perturbation that preserves the symmetry $G$ but splits the energy level, which contradicts the idea that the degeneracy is a fundamental consequence of the symmetry.

This leads to the central conclusion of this lesson:

The set of eigenstates for a given energy level forms a basis for an irreducible representation (irrep) of the system's symmetry group. The degeneracy of the energy level is therefore equal to the dimension of that irreducible representation.

This insight is immensely powerful. If we know the symmetry group of a Hamiltonian, we can immediately classify its possible energy degeneracies by simply listing the dimensions of the group's irreps. For example, if a group has irreps of dimensions 1, 1, and 2, we should expect to find non-degenerate energy levels (singlets) and two-fold degenerate levels (doublets) in the spectrum.

This is formalized in the following resource.

3. The Algebraic Origin of Degeneracy

The link between group properties and degeneracy becomes clearer when we look at continuous symmetries and their generators. As you know from quantum mechanics, continuous symmetries (like rotations) are associated with conserved quantities (like angular momentum) which are the generators of the symmetry transformations.

Let's consider two symmetry operators, $U(g_1)$ and $U(g_2)$. If they do not commute, the group is non-abelian. Their generators, say $J_1$ and $J_2$, will also fail to commute. It is this non-commutativity that lies at the heart of symmetry-induced degeneracy.

If a Hamiltonian has a non-abelian symmetry group, its spectrum must have degenerate levels. If all energy levels were non-degenerate, the eigenstates of $H$ would form a unique basis. Since all generators $J_i$ commute with $H$, they would all be diagonal in this same basis, which would mean they all commute with each other—a contradiction.

The following lecture by Leonard Susskind provides a wonderful, physics-first development of these ideas.

This final point about angular momentum is a cornerstone example. The set of $2j+1$ states $\{|j,m⟩:m=-j,\dots ,+j\}$ that are connected by the raising and lowering operators form a basis for a $(2j+1)$-dimensional irreducible representation of the rotation group SO(3). Because the Hamiltonian of a system with rotational symmetry (like the hydrogen atom) commutes with all the angular momentum operators, all these $2j+1$ states must be degenerate.

4. Accidental Degeneracy

Not all degeneracies are a direct consequence of a system's symmetry group. Sometimes, for a specific set of parameters in the Hamiltonian (e.g., specific values for masses, coupling constants, or field strengths), energy levels belonging to different irreducible representations might happen to coincide. This is called accidental degeneracy.

The key difference is that accidental degeneracies are not "protected" by symmetry. A small change in the Hamiltonian's parameters that still respects the overall symmetry of the system can lift an accidental degeneracy. In contrast, an "essential" degeneracy—one corresponding to a multi-dimensional irrep—cannot be lifted without breaking the symmetry of the system itself.

Your background in condensed matter physics makes tight-binding models an excellent context for exploring this. Consider the following example.

This example with the C60 molecule is particularly illustrative. The icosahedral symmetry $I_h$ of the molecule dictates that there must be degeneracies of dimension 1, 3, 4, and 5, corresponding to the dimensions of the group's irreps. However, for a simple model where all hopping parameters are equal, a 9-fold degenerate level appears. This does not match any irrep dimension and is an accidental degeneracy between a 4-fold and a 5-fold degenerate multiplet. Making the model more realistic by distinguishing between two inequivalent types of bonds—a change that preserves the full $I_h$ symmetry—lifts this accidental degeneracy while preserving the essential ones.

Conclusion

In this lesson, we have forged the crucial link between the abstract definition of a symmetry group and the concrete, observable structure of a quantum system's energy spectrum.

Key Takeaways:

• If a Hamiltonian $H$ is symmetric under a group $G$, it commutes with all the group's representation operators: $[H,U(g)]=0$.
• This commutation implies that the eigenspace corresponding to a specific energy level is closed under the group's action and thus carries a representation of $G$.
• In general, this representation is irreducible, and the degeneracy of the energy level is equal to the dimension of the irrep. These are known as essential degeneracies.
• The non-commutativity of generators in a non-abelian group is the algebraic origin of multi-dimensional irreps and, consequently, of essential degeneracies.
• Accidental degeneracies are coincidences of energy levels from different irreps, which occur for specific parameter values in the Hamiltonian and are not protected by the group symmetry.

You now understand not just that energy levels are degenerate, but why they are, and how to classify them using the language of group theory. This principle is a cornerstone of spectroscopy, condensed matter physics, and particle physics.

In our next lesson, we will examine the other side of this coin. We've seen how symmetry constrains the static properties of the Hamiltonian (its spectrum). Next, we will see how symmetry constrains the dynamics of the system by giving rise to conservation laws, a principle encapsulated in Noether's theorem.