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In this note I give some particular cases of the module <tt>NormalOrder</tt> (see [[Blog:Notes/NormalOrder module|here]] for this module itself). | In this note I give some particular cases of the module <tt>NormalOrder</tt> (see [[Blog:Notes/NormalOrder module|here]] for this module itself). | ||
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--more-- | --more-- | ||
| − | |||
| − | :< | + | If <math>C</math> commutes with <math>A</math> and <math>B</math> then: |
| + | |||
| + | :<math>[A,BC]=[A,B]C</math> | ||
So it is easy to compute expressions like, e.g., | So it is easy to compute expressions like, e.g., | ||
| − | :< | + | :<math>[a^\dagger a,a^\dagger b]</math> |
factoring out b and computing: | factoring out b and computing: | ||
| Line 17: | Line 19: | ||
we find: | we find: | ||
| − | :< | + | :<math>[a^\dagger a,a^\dagger b]=a^\dagger b\,.</math> |
| − | If you followed my trend yesterday you see we're getting closer to a universal engine to solve symbolically (or reduce) quantum systems. | + | 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 | + | Here is a module that does the commutation directly, computing <math>:[elem_1,elem_2]:</math>: |
<pre> | <pre> | ||
| Line 49: | Line 51: | ||
</pre> | </pre> | ||
| − | already given above, or | + | which is the case already given above, or |
<pre> | <pre> | ||
| Line 56: | Line 58: | ||
</pre> | </pre> | ||
| − | which is the fundamental relation < | + | which is the fundamental relation <math>[a,a^\dagger]=1</math>. |
| − | {{wl-publish: 2010-11-03 14:08:36 - | + | {{wl-publish: 2010-11-03 14:08:36 -0500 | Fabrice }} |
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 \([a,a^\dagger]=1\).