The general Schrödinger equation. x, p commutator
PROFESSOR: ih bar d psi dt equal E psi where E hat is equal to p squared over 2m, the operator. That is the Schrodinger equation. The free particle Schrodinger equation-- you should realize it's the same thing as this. Because p is h bar over i ddx. And now Schrodinger did the kind of obvious thing to do. He said, well, suppose I have a particle moving in a potential, a potential V of x and t-- potential. Then the total energy is kinetic energy plus potential energy. So how about if we think of the total energy operator. And here is a guess. We'll put the just p squared over 2m, what we had before. That's the kinetic energy of a particle. But now add plus V of x and t, the potential. That is reasonable from your classical intuition. The total energy is the sum of them. But it's going to change the Schrodinger equation quite substantially. Now, most people, instead of calling this the energy operator, which is a good name, have decided to call this the Hami...