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## The representation theory of SU(2)

Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of $\text{SU}(2)$ (which you’ll recall we assumed way back in this previous post). As a first step, it turns out that the finite-dimensional representation theory of compact groups looks a lot like the finite-dimensional representation theory of finite groups, and this will be a major boon to three of the proofs. The last proof will instead proceed by classifying irreducible representations of the Lie algebra $\mathfrak{su}(2)$.

At the end of the post we’ll briefly describe what we can conclude from all this about electrons orbiting a hydrogen atom.

## More about the Schrödinger equation on a finite graph

It looks like the finite graph model is not just a toy model! It’s called a continuous-time quantum random walk and is used in quantum computing in a way similar to how random walks on graphs are used in classical computing. The fact that quantum random walks mix sooner than classical random walks relates to the fact that certain quantum algorithms are faster than their classical counterparts.

I learned this from a paper by Lin, Lippner, and Yau, Quantum tunneling on graphs, that was just posted on the arXiv; apparently the idea goes back to a 1998 paper. I have an idea about another sense in which the finite graph model is not just a toy model, but I have not yet had time to work out the details.