In this note I give some particular cases of the module NormalOrder (see here for this module itself).
If \(C\) commutes with \(A\) and \(B\) then:
\[[A,BC]=[A,B]C\]
So it is easy to compute expressions like, e.g.,
\[[a^\dagger a,a^\dagger b]\]
factoring out b and computing:
NormalOrder[{{{1, 1, 1, 0}, 1}, {{2, 1}, -1}}]
{{{1, 0}, 1}}
we find:
\[[a^\dagger a,a^\dagger b]=a^\dagger b\,.\]
If you followed my trend yesterday you see we're getting closer and closer to a universal engine to solve symbolically (or reduce) quantum systems.
Here is a module that does the commutation directly, computing \(:[elem_1,elem_2]:\):
NormalOrderCommutator[elem1_, elem2_] := Module[{},
NormalOrder[{prod[elem1, elem2], prod[{1, -1}*elem2, elem1]}]
]
in term of another module, prod, which is a component of the NormalOrder module:
prod[sequence__] := Module[{terms},
terms = Transpose[List[sequence]];
{Associate[Flatten[EvenSize /@ (terms[[1]])]], Times @@ terms[[2]]}
]
EvenSize[list_] := Module[{},
If[OddQ[Length[list]], Prepend[list, 0], list]
]
Some examples:
NormalOrderCommutator[{{1, 1}, 1}, {{1, 0}, 1}]
{{{1, 0}, 1}}
which is the case already given above, or
NormalOrderCommutator[{{1}, 1}, {{1, 0}, 1}]
{{{0, 0}, 1}}
which is the fundamental relation \([a,a^\dagger]=1\).