John Carlos Baez on Nostr: I've been talking about how a classical particle in an inverse square force law, like ...

I've been talking about how a classical particle in an inverse square force law, like the Earth orbiting the Sun, can secretly be seen as a particle moving at the speed of light on a sphere in 4-dimensional space.

That's already cool. But the same story also works in quantum mechanics, like for an electron orbiting a proton in a hydrogen atom!

There are various ways to show this is true. I'll take a brutally efficient approach. I’ll start with wavefunctions on a sphere in 4d space and describe operators on these arising from rotational symmetries. Combining these with standard facts about the hydrogen atom, we can set up a correspondence between such wavefunctions and bound states of a hydrogen atom — that is, states where the electron orbits the proton, instead of flying off.

Here’s how. Chemists use a basis of bound states of the hydrogen atom called |𝑛,ℓ,𝑚〉where

• n tells us the energy of the state,
• ℓ tells us its total angular momentum, and
• m tells us its angular momentum around the z axis.

(For now we’re ignoring the electron’s spin: I'll bring that in later.) In this blog article, I'll show that wavefunctions on the sphere in 4-dimensional space *also* have a nice basis of states called |𝑛,ℓ,𝑚〉. This will let us match up hydrogen atom bound states and wavefunctions on a sphere in 4d space.

So the most basic element turns out to be 4-dimensional, in a sense.

https://johncarlosbaez.wordpress.com/2025/07/23/the-kepler-problem-part-5/