Lecture 6 8.321 Quantum Theory I, Fall 2017 33
Lecture 6 (Sep. 25, 2017)
6.1 Solving Problems in Convenient Bases Last time we talked about the momentum and position bases. From a practical point of view, it often pays off to not solve a particular problem in the position basis, and instead choose a more convenient basis, such as the momentum basis. Suppose we have a Hamiltonian
1 d2 H = − + ax . (6.1) 2m dx2 We could solve this in the position basis; it is a second-order differential equation, with solutions called Airy functions. However, this is easy to solve in the momentum basis. In the momentum basis, the Hamiltonian becomes p2 d H = + ia , (6.2) 2m ~dp which is only a first-order differential equation. For other problems, the most convenient basis may not be the position or the momentum basis, but rather some mixed basis. It is worth becoming comfortable with change of basis in order to simplify problems.
6.2 Quantum Dynamics Recall the fourth postulate: time evolution is a map