Open course ware

BARTON ZWIEBACH: Hydrogen atom is the beginning of our analysis. It still won't solve differential equations, but we will now two particles, a proton, whose coordinates are going to be coordinate of the proton, subbing Xp for the proton, and momentum of the proton. And there's an electron. And there's the coordinates, the three coordinates of the electron, and the three components of the momenta of the electron. And these are your canonical variables. This means that the components of this object satisfy the standard commutation relations. That is-- I have to write it like the following. They considered the coordinates of the proton, the i-th component. And the momentum of the proton, the j-th component, that's equal to ih bar delta ij. You see, we used to code for its x, y, and z, and momenta Px, Py, Pz. You could've called it X1, X2, X3, momenta P1, P2, P3. And in that way, you can use a Kronecker delta over here. So, these are the commutation relati...

PROFESSOR: We have the hydrogen atom Hamiltonian. Hamiltonian. And that was given by the kinetic operator for the proton plus the kinetic operator for the electron plus the potential, which was a function of the distance between the proton and the electron. And what we achieved last time was the introduction of two new pairs of canonical variables. We had the electron position momentum, that's a pair of canonical variables. The proton position and momentum, that's another pair of canonical variables. They commute each pair, the two operators commute to give IH bar, but the two pairs are independent. So we search for another two pairs of variables, and we found another two pairs. One was the P and X associated with the center of mass motion, and then we had the small p and small x associated with the relative function. And these four variables were a function of the original four variables, the X and P of the electron and the x and p of the proton. So we define the...