Quantum mechanics meets the dodecahedron

Yet another great image by +Greg Egan. I'll try to explain it in simple terms.

In quantum mechanics, the position of a particle is not a definite thing: it's described by a wavefunction. This says how probable it is to find the particle at any location... but it also contains other information, like how probable it is to find the particle moving at any velocity.

Take a hydrogen atom, and look at the wavefunction of the electron.

Puzzle 1. Can we make the electron's wavefunction have all the rotational symmetries of a dodecahedron - that wonderful Platonic solid with 12 pentagonal faces?

Yes! In fact it's too easy: you can make the wavefunction look like whatever you want.

So let's make the puzzle harder. Like everything else in quantum mechanics, angular momentum can be uncertain. In fact you can never make all 3 components of angular momentum take definite values! However, there are lots of wavefunctions where the magnitude of the angular momentum is completely definite.

Puzzle 2. Can an electron's wavefunction have a definite magnitude for its angular momentum while having all the rotational symmetries of a dodecahedron?

Yes! And there are infinitely many ways for this to happen! +Greg Egan drew the simplest one here:


and this started a long discussion. By the end, we had completely crushed the problem. So, we could solve harder puzzles.

The magnitude of the angular momentum is determined by a number called ℓ, for some idiotic reason. And this number is quantized! It can only take values 0, 1, 2, 3, ... and so on.

The simplest solution to Puzzle 2 has ℓ = 6, for some reason that's not at all idiotic. We can get it using the 6 lines connecting opposite faces of the dodecahedron!

How does that work? Well, read the discussion on Egan's post. It takes some math to see how it works.

Puzzle 3. What's the smallest choice of ℓ where we can find two different electron wavefunctions that both have the same ℓ and both have all the rotational symmetries of a dodecahedron?

It turns out to be ℓ = 30. The picture on this post shows a wavefunction oscillating between these two possibilities!

But we can go a lot further:

Puzzle 4. For each ℓ, how many linearly independent electron wavefunctions have that value of ℓ and all the rotational symmetries of a dodecahedron?

For ℓ ranging from 0 to 29 there are either none or one. There are none for these numbers:

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29

and one for these numbers:

0, 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28

(I said the case ℓ = 6 gave the simplest wavefunction with dodecahedral symmetry, but I was lying. The case ℓ = 0 gives a constant wavefunction. This has dodecahedral symmetry, but it's completely boring: the picture would be a featureless sphere!)

The pattern continues as follows. For ℓ ranging from 30 to 59 there are either one or two. There is one for these numbers:

31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 53, 59

and two for these numbers:

30, 36, 40, 42, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58

The numbers in these lists are just 30 more than the numbers in the first two lists! And it continues on like this forever.

Puzzle 5. What's special about these numbers from 0 to 29?

0, 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28

You don't need to know tons of math to figure this out - but I guess it's a sort of weird pattern-recognition puzzle unless you know the math that says which patterns are likely to be important here. So, as a hint, I'll say that writing numbers as sums of other numbers is important.

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