Massachusetts Institute of Technology
Department of Physics
Physics 8.04 Thu Oct 12 16:08:30 EDT 1995
So far we have built the following items into our theory: quantum states and superposition, the interpretation of the magnitude squared of quantum amplitudes as probabilities, both de Broglie Hypothesis and the notion of quantum determinism.
As partial verification, out of our theory we have recovered the following new physical facts which were not built directly into the theory, the Heisenberg uncertainty principle, the conservation of probabilities and the correspondence principle (Ehrenfest's theorem).
We now enter the next phase of the course where we will use our formalism to predict new phenomena. To make these predictions, in principle all that one needs to know is the time dependent Schrödinger equation (TDSE) describing how our quantum states change in time and how to use the resulting wavefunctions to compute quantum probabilities and averages. All of the development you will see throughout the rest of this course follows directly from the TDSE.
However, the TDSE is a complicated partial differential equation. While (as in Problem Set 6) one may at times be able to ``guess'' at solutions to the Schrödinger equation, it is necessary to equip ourselves with a set of general mathematical tools for solving the TDSE. The purpose of this note is to give you the first tool in this arsenal, the method of separation of variables. With separation of variables you will be able to eliminate the time dependence from the TDSE and produce a new, time independent equation, the Time Independent Schrödinger Equation (TISE). For one dimensional systems, the TISE will no longer be a partial differential equation but will be an ordinary differential equation which is then much easier to solve.
After deriving the TISE, this note will go on to explore some of the general features of its solutions. We do this before we move on in the course to discuss methods to solve the TISE in specific instances. The first methods we will discuss for solving the TISE in one dimension will be the qualitative methods laid out in Section 3.11 of the text, French and Taylor.