Line 1: | Line 1: | ||
− | In this note I give some particular cases of the module <tt>NormalOrder</tt> (see [[Blog:Notes/NormalOrder module|here]]). | + | In this note I give some particular cases of the module <tt>NormalOrder</tt> (see [[Blog:Notes/NormalOrder module|here]] for this module itself). |
If <m>C</m> commute with <m>A</m> and <m>B</m> then: | If <m>C</m> commute with <m>A</m> and <m>B</m> then: | ||
Line 18: | Line 18: | ||
:<m>[a^\dagger a,a^\dagger b]=a^\dagger b\,.</m> | :<m>[a^\dagger a,a^\dagger b]=a^\dagger b\,.</m> | ||
+ | |||
+ | If you followed my trend yesterday you see we're getting closer to a universal engine to solve symbolically (or reduce) quantum systems. | ||
Here is a module that does it directly, computing <m>:[elem_1,elem_2]:</m>: | Here is a module that does it directly, computing <m>:[elem_1,elem_2]:</m>: | ||
Line 55: | Line 57: | ||
which is the fundamental relation <m>[a,a^\dagger]=1</m>. | which is the fundamental relation <m>[a,a^\dagger]=1</m>. | ||
+ | {{wl-publish: 2010-11-03 14:08:36 -0400 | Fabrice }} |
In this note I give some particular cases of the module NormalOrder (see here for this module itself).
If <m>C</m> commute with <m>A</m> and <m>B</m> then:
So it is easy to compute expressions like, e.g.,
factoring out b and computing:
NormalOrder[{{{1, 1, 1, 0}, 1}, {{2, 1}, -1}}] {{{1, 0}, 1}}
we find:
If you followed my trend yesterday you see we're getting closer to a universal engine to solve symbolically (or reduce) quantum systems.
Here is a module that does it directly, computing <m>:[elem_1,elem_2]:</m>:
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}}
already given above, or
NormalOrderCommutator[{{1}, 1}, {{1, 0}, 1}] {{{0, 0}, 1}}
which is the fundamental relation <m>[a,a^\dagger]=1</m>.