In this episode we see one of the reasons why complex numbers are useful in Quantum Mechanics. This isn’t the only reason why they are useful. We’ll see another one when we look at continuous probability densities.
It’s also not the only way of overcoming certain problems. QM can be formulated in the language of Clifford Algebras. (See for example “Geometric Algebra for Physicists” by Doran and Lasenby.) Since Clifford Algebras encompass complex numbers, this isn’t really that surprising.
Next week, we’ll see what our first complex operator looks like and will find that we’ve actually discovered the Pauli spin matrices.