# NormalOrder examples

⇠ Back to Blog:Notes

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 $$[a,a^\dagger]=1$$.