Algebraic solution of the harmonic oscillator
PROFESSOR: SHO algebraically. And we go back to the Hamiltonian, p squared over 2m plus 1/2 m omega squared x hat squared. And what we do is observe that this some sort of sum of squares plus p squared over m-- p squared over m squared omega squared. So the sum of two things squared. Now, the idea that we have now is to try to vectorize the Hamiltonian. And what we call vectorizing is when you write your Hamiltonian as the product of two vectors, V times W. Well actually, that's not quite the vectorization. You want kind of the same vector, and not even that. You sort of want this to be the Hermitian conjugate of that. And if there is a number here, that's OK. Adding numbers to a Hamiltonian doesn't change the problem at all. The energies are all shifted, and it's just how you're defining the zero of your potential, is doing nothing but that. So vectorizing the Hamiltonian is writing it in this way, as V dagger V. And you would say, why V dagger V? Why...