The simplest quantum dodecahedron

Thanks to a suggestion from +Gerard Westendorp, I’ve been looking at wave functions with the symmetry group of the icosahedron / dodecahedron. At first I thought such things couldn’t exist at all, but there are actually some very simple examples.

One way to construct such a wave function is to pick a certain value of the quantum number, ℓ, which describes the total angular momentum of the particle. This choice of ℓ also fixes a certain irreducible representation of the group of all three-dimensional rotations, SO(3), on a space of wave functions of dimension d=2ℓ+1. (Here we are ignoring the radial part of the wave function, and just concentrating on its angular distribution, which is what matters in terms of the rotational symmetry properties we are interested in.)

That space of wave functions is just the space of spherical harmonics Y_{ℓ,m} with our chosen value of ℓ, and the 2ℓ+1 possible values of m that range from –ℓ to ℓ. Here, m is the quantum number that gives the component of the angular momentum measured along the z-axis in our chosen Cartesian coordinate system.

Associated with any three-dimensional rotation, R, is a unitary matrix ρ_ℓ(R) that acts on the space of spherical harmonics. If we want to create a wave function that is invariant under any rotation in the icosahedral symmetry group, G, we can define the matrix:

T = sum over all R in G of ρ_ℓ(R)

and then, starting from (almost) any wave function we like, say ψ_0, we compute:

ψ = T ψ_0

If we act on ψ with ρ_ℓ(S) for any rotation S in the symmetry group G, we have, for each ρ_ℓ(R) that is summed to make T:

ρ_ℓ(S) ρ_ℓ(R) = ρ_ℓ(SR)

This is true by the definition of ρ_ℓ being a representation of the group of all rotations. But since we are summing over every R in G, the new rotations, SR, will just be a different way of writing all the rotations in G in a different order. G is a group, so it closed under multiplication.

There are two things that could go wrong with this construction. One is that T might turn out to be the zero matrix, which is fatal. The other is that we might choose ψ_0 so that ψ = T ψ_0 is zero, even though T itself is non-zero. In that case, we really just need to look more carefully at T, and choose ψ_0 more wisely.

It turns out that the first few values of ℓ for which T is non-zero are:

ℓ = 6, 10, 12, 15, 16, 18, 20.

In all of these cases, T has rank 1, which means there is just a 1-dimensional space of invariant vectors that we can obtain from it. So if we normalise the invariant vector, the result will be unique up to an overall phase.

For ℓ = 6, the invariant vector has a remarkably simple form in the standard basis:

ψ = (√7 |m=–5> + √11 |m=0> – √7 |m=5>)/5

where we are using “ket” notation for the wave functions, and we are taking ℓ = 6 as given in all these kets. If you prefer to see this written in terms of spherical harmonics:

ψ = (√7 Y_{6,–5} + √11 Y_{6,0} – √7 Y_{6,5})/5

So, using just three eigenstates for the component of angular momentum along the z-axis, we can construct a wave function with perfect dodecahedral symmetry!
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