Commutators and Bra-Ket Notation in Quantum Mechanics

## MathCAT Version

MathCAT-0.6.6 (NVDA Add-On)

## Description of Issue

### Commutators

MathCAT treats `$[A, B]$` and other expressions of that form as intervals, but such expressions represent commutators in quantum mechanics. For instance, MathCAT reads $[A, B]$ as "the interval from cap A to cap B, including cap A and cap B." For nested commutators, however, it becomes particularly hard to parse. For the commutator of A with the commutator of B with C ($[A, [B, C]]$), MathCAT reads, "the interval from cap A to the interval from cap B to cap C, including cap B and cap C, including cap A and the interval from cap B to cap C, including cap B and cap C."

### Bra-Ket Notation

For bra-ket notation, MathCAT reasonably verbalizes the ket of alpha ($| \alpha \rangle$) as "vertical line alpha right angle bracket" and the bra of alpha ($\langle \alpha |$) as "left angle bracket alpha vertical line." For expectation values, howeover, MathCAT verbalizes $\langle \alpha | \hat{P} | \alpha \rangle$ as "left angle bracket, alpha, the absolute value of cap P hat, end absolute value, alpha, right angle bracket." Likewise, the inner product of the bra of alpha and the ket of beta ($\langle \alpha | \beta \rangle$) is verbalized as "left angle bracket, alpha divides beta, right angle bracket." (In those expressions where the vertical bar is verbalized as "divides," the symbol is treated as a symbol of relation and is surrounded by spaces in Nemeth.) In some equations that relate expressions of kets, the inferred "absolute value" extends across the equals sign, from one vertical bar to the next, and this is particularly confusing to parse.

## Potential Resolution

Perhaps it is best not to infer that bracketed expressions of the form `$[A, B]$` are instances of interval notation. It is not clear, without intent markup by the author, how MathCAT would be able to correctly verbalize bra-ket notation.

In either case, it seems that literal verbalization (issue #135) would be a good feature to offer, since it would leave interpretation to the user (as is the case for visual and braille interpretation).

The [MathCAT User Guide](https://nsoiffer.github.io/MathCAT/users.html) states, "Expressions are never unambiguous in SimpleSpeak." It would seem that ClearSpeak is thus recommended for mathematicians, physicists, etc., but the inferences made by ClearSpeak can make interpreting certain types of expressions harder than interpreting literal verbalization of symbols.