Further changes to representations

This past week brought further changes to how representations are done in sympy.physics.quantum, as well as some basic tests and doctests.

First, I added some basic tests for the default label functionality that I talked about last week. Those are here.

Beyond that, I added code to represent which would allow for calculation of operator representations, in the form of <x’ | A | x>. There is still a question, which was discussed over the list, of how to deal with representations of arbitrary expressions. I think the idea is simply to try to apply a bra from the basis you are attempting to represent in on the left of the expression. One small subtlety is that this would mean that the calling represent with the position basis on A | x> would give you the same result as calling represent on just A (that is, <x’ | A | x>). The code for that, and tests, is in this commit.

Next, I set about making some basic Wavefunction and DifferentialOperator classes. There was some debate as to whether Wavefunction is needed at all, and I think there are some justifications for it. First of all, it would mean that all representations in continuous bases could have the same class Rather than returning simple expressions, a function, which could be evaluated at any point would be returned. Additionally, this class can have extra convenience functions for calculating normalization constants or calculating quantum probabilities. It could also internally handle things like indexing of states. Finally, it makes writing DifferentialOperator a bit easier. In fact, it is as simple as implementing an _apply_operator_Wavefunction method, to calculate the derivative. There has still been debate over the implementation of Wavefunction, whether it should inherit from Lambda or Function. I have been trying to sort out implementation issues like this, because Function itself is not callable so I need to define my own __call__ method, which essentially duplicates functionality that already exists in Lambda. Initial code for this is here, with the first tests here. Some changes that I made today after reading discussions on the list, are here.

In the next week I’d really like to nail down the implementation of Wavefunction, as well as an example using the already written code in piab.py and changing represent to use Wavefunction. Hopefully by the end of the week I can have a nice example and some pretty plots!