The commutator is useful when you want to switch the order of acting with operators. It is defined [A,B] as the difference between acting on a function with A then B and the operation of acting on a function with B then A. If the commutator equals zero, then the operations A and B commute, meaning you can do them in either order it doesn't matter. If the commutator is non-zero, you can still switch the order of AB to BA, but you need to add the commutator. AB f(x) = BA f(x) + [A,B] f(x)
For two quantities to be simultaneously observable, the commutator must equal zero. (By wizardry of the Quantum Mechanics gods, or higher math that requires hand-waving.) So the fact that the position and momentum operator do not commute is a manifestation of the uncertainty principle, that position and velocity of a 'particle' cannot be simultaneously known to arbitrary precision.
'a' and 'a dagger' operators are the raising and lowing (or creation and annihilation) operators, respectively. It's cleaner notation-wise to do computations with these operator symbols for ideal systems with nice analytic solutions like the harmonic oscillator. They increase or decrease the energy of the state by hw. Or they can be thought of as adding or subtracting a particle in an energy state.
'a dagger' is the Hermitian adjoint of 'a'. That [a, a dagger] = 1 makes it a unitary operator. What all these properties, do I don't know. I suspect they create useful superpowers for the operations.
'H' is the Hamiltonian operator in the Schrodinger equation.
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