The Kepler Problem (Part 9)

Today I want to make a little digression into the quaternions. We won’t need this for anything later—it’s just for fun. But it’s quite beautiful.

We saw in Part 8 that if we take the spin of the electron into account, we can think of bound states of the hydrogen atom as spinor-valued functions on the 3-sphere. Here a ‘spinor’ is a pair of complex numbers.

But we can also think of a spinor as a quaternion! And we can think of the 3-sphere as the unit sphere in the quaternions! So bound sta...