expand noncommutative products over sums, Commutators, Brackets, etc. - Maple Help

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Physics[Expand] - expand noncommutative products over sums, Commutators, Brackets, etc.

Calling Sequence

expand(expr)

Parameters

expr

-

any expression

Description

• 

The Expand command expands powers and distributes products over sums, where the products can be expressed with the * and ^ operators in the Physics package, or with the commutative (:-*, :-^) or inert (%*, %^) versions of them. Expand also expands the Commutator and AntiCommutator functions of the Physics package, Brackets and Inverse of products. The products resulting from these expansions are all returned normalized using Normal.

• 

After the Physics package has been loaded, the functionality provided by Expand is also automatically provided through the standard expand command. However expand additionally expands mathematical functions, something that Expand does not do.

  

Note: For the conventions adopted to represent noncommutative and anticommutative objects, see Setup and the types anticommutative and noncommutative.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

For illustration purposes, first set theta and Q as prefixes to identify anticommutative variables and functions, and Z to identify noncommutative ones (see Setup for details).

Setupanticommutativeprefix=Q,θ,noncommutativeprefix=Z

anticommutativeprefix=Q,_λ,θ,noncommutativeprefix=Z

(2)

Consider now the noncommutative product between anticommutative objects and related sums.

Q1Q1+Q3Q4+Q5Q6

Q1Q1+Q3Q4+Q5Q6

(3)

You can get the expanded form of this product using expand or Expand:

Expand

Q1Q3Q4Q6+Q1Q3Q5Q6

(4)

Note that in the expanded representation, all * products are distributed. This is a more complicated example, involving anticommutative and noncommutative objects.

ahθ+bQ1Z1Q2Q4+Z3Q5cQ3Q6fθ

ahθ+bQ1Q2Q4Z1+cfθQ5Q3Q6Z3

(5)

Expand

ahθQ1Q2Q4Z1+abQ1Q2Q4Z1cfθQ3Q5Z3cfθQ5Q6Z3

(6)

Now you can use the usual Maple commands to manipulate this expression. For example, note the existence of common factors entering the commutative products of this expression; you can take advantage of them to simplify it.

convert,horner

cfθQ3Q5Z3fθQ5Q6Z3+ahθQ1Q2Q4Z1+bQ1Q2Q4Z1

(7)

To additionally expand also the mathematical functions, use expand instead of Expand; compare for instance these two results:

Z3Q5cQ3Q6cosa+b

ccosa+bQ5Q3Q6Z3

(8)

Expand

ccosa+bQ3Q5Z3ccosa+bQ5Q6Z3

(9)

expand

ccosacosbQ3Q5Z3ccosacosbQ5Q6Z3+csinasinbQ3Q5Z3+csinasinbQ5Q6Z3

(10)

Expansion of Brackets over sums happen automatically but for inert Brackets you can use Expand

Setupquantumoperators=A,B

quantumoperators=A,B

(11)

%BracketBraφ,aA+bB+c,Ketψ

φ|aA+bB+c|ψ

(12)

Expand

aφ|A|ψ+bφ|B|ψ+cφ|ψ

(13)

See Also

convert/horner, Physics, Physics conventions, Physics examples, Physics[*], Physics[^], Setup


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