Responsible party: East End, Devlin

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Please try to include the following

• main points understood, and expand them - what is your understanding of what the points were.
  • expand these points by including many of the details the class discussed.
• main points which are not clear. - describe what you have understood and what the remain questions surrounding the point(s).
  • Other classmates can step in and clarify the points, and expand them.
• How the main points fit with the big picture of QM. Or what is not clear about how today's points fit in in a big picture.
• wonderful tricks which were used in the lecture.

Steps to the Analytic Method:

1) Use dimensionless form of DE

<math>let \xi = \sqr [(mx\omega}/\hbar)] </math> and <math> K=(2E)/(\hbar\omega). </math>

Then we can use the dimensionless form of the Schrodinger <math> \frac{\partial^2}{\partial x^2}=(\xi^2-K)\psi(x) </math>

We can think of <math> \xi </math> as approximately <math> x </math> and also <math> \psi </math> as approximately <math> e^(-(\xi)^2/2)</math>.

2.) <math> \psi\approx h(\xi)e^(-\xi^2/2) </math>

We use this and hope that <math> h(\xi) </math> is simpler than <math> \psi(\xi) </math>

3.) Substitute <math> \psi </math> into <math> \frac{\partial^2}{\partial x^2}=(\xi^2-K)\psi(x). </math>

Differentiate and then Schrodinger's equation becomes

<math> \frac{\partial^2h(\xi)}{\partial \xi^2}=-2\xih(\xi)+(K-1)(h(\xi)=0 </math> (A)

4.) Use power series to find a solution.

<math> h(\xi)=\sum a\sub n * \xi^n. </math>.

Differentiate each term twice and then plug that into (A) and we get a recursive equation that can be illustrated like this:

<math> (blah)(\xi)^0+ (blahblah)(\xi)^1 +(moreblah)(\xi)^2+…=0. </math>

Since the equation needs to hold true for all <math> \xi </math>, the blahs must equal zero. We now have this equation

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