If you have never used Mathematica before, you might take twenty minutes and take this tutorial. Also, Wolfram Research has released two Hands-on Start to Mathematica, Hands-on Start to Mathematica--Part 2.
Your best bet for quickly finding specific information on a function is to open the documentation center (Help → "Documentation Center"). Type what you want to do into the search bar. It helps if you know the function name to avoid getting lots of hits which are not quite what you want (e.g., if you type "Graph" instead of "Plot", you will get lots of Graph functions which are probably not what you want although Plot will show up after a few pages).
One of the cool things about the documentation center is that you can try thing out. Modify an example, hit "Shift-Enter" and see the new result.
Here is a notebook which has tips for Integration & functions. Download the link and open it in Mathematica.
Introduction to Plot and Manipulate
In quantum mechanics, we use a lot of complex functions so here are some tips on using Complex Functions in Mathematica
While Mathematica excels at symbolic manipulation, it is also very adept at finding numerical solutions to problems where an analytic solution is not possible. Here is a simple example.
Mathematica can show you what a wave function looks like as it evolves in the complex plane. Here is a notebook which shows the time dependence of eigenstates of the infinite square well.
Here is a notebook which shows how we can match an initial wave function which is not an energy eigenstate and then see how this match evolves in time.
Need a quick page to search through for some functions and examples? Well Prof. Richard Sorbello from the University of Wisconsin at Milwaukee has put together a LIST OF MATHEMATICA COMMANDS AND EXAMPLES.
For a more thorough introduction including tips and tricks on formatting and dealing with errors, see Prof. Herbert Halpern's General Mathematica Instructions (pdf from the University of Cincinnati).
Physics 3220 home page.
This page maintained by Steve Goldhaber.