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PROFESSOR: This is a wonderful differential equation, because it carries a lot of information. If you put this psi, its certainly going to be a solution. But more than that, its going to tell you the relation between k and omega. So if you try your-- we seem to have gone around in circles. But youve obtained something very nice. First, we claim that thats a solution of that equation and has the deep information about it. So if you try again, psi equal e to the ikx minus i omega t, what do we get? On the left hand side, we get ih minus i omega psi. And on the right hand side, we get minus h squared over 2m and two derivatives with respect to x. And that gives you an ik times and other ik. So ik squared times psi. And the psis cancel from the two sides of the equation. And what do we get here? h bar omega. Its equal to h squared k squared over 2m, which is e equal p squared over 2m. So it does the whole job for you. That differential equation is quite smart. It admits these as solu
