Download Pauli Pascal Triangle and other Noncommutative Expansions

Pauli Pascal Triangle and other Noncommutative
Expansions
by José Luis Gómez-Muñoz
http://homepage.cem.itesm.mx/lgomez/quantum/
[email protected]
With contributions by Gianfranco Arroyo-Orrico and Sergio E. Martinez-Casas
Introduction
This is document shows algebraic expansions of binomials for operators with different commutation or anticommutation
relationships.
Standard Pascal Triangle for Scalars
First we show the standard Pascal Triangle for commutative algebra:
Clear@a, bD;
TraditionalForm@Grid@
Table@8
Expand@Ha + bLn D
<, 8n, 1, 7, 1<D,
Dividers → AllDD
a+b
a2 + 2 a b + b2
3
a + 3 a2 b + 3 a b2 + b3
a4 + 4 a3 b + 6 a2 b2 + 4 a b3 + b4
5
a6
a + 5 a4 b + 10 a3 b2 + 10 a2 b3 + 5 a b4 + b5
+ 6 a5 b + 15 a4 b2 + 20 a3 b3 + 15 a2 b4 + 6 a b5 + b6
a7 + 7 a6 b + 21 a5 b2 + 35 a4 b3 + 35 a3 b4 + 21 a2 b5 + 7 a b6 + b7
Coefficients of the standard Pascal Triangle:
2
v7triangles.nb
GridA
TableA9
expanded = Expand@Ha + bLn D;
TableACoefficientAexpanded, an−j bj E, 8j, 0, n<E
=, 8n, 1, 7, 1<E,
Dividers → AllE
81, 1<
81, 2, 1<
81, 3, 3, 1<
81, 4, 6, 4, 1<
81, 5, 10, 10, 5, 1<
81, 6, 15, 20, 15, 6, 1<
81, 7, 21, 35, 35, 21, 7, 1<
In this Pascal triangle each new number is the addition of the two numbers above it.
Pauli-Pascal Triangle for Anticommuting Operators
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Press [ESC]anti[ESC] in order to show the anticommutator template (notice the + symbol in the subscript).
Here we assign the value of zero to the anticommutator of these two operators:
Clear@x, yD;
SetQuantumObject@x, yD;
Px, yT+ = 0
0
Anticommutating operators generate the Pauli-Pascal Triangle. However, it does not look like a triangle, because the coefficients equal to zero are not shown. Notice that "CommutatorExpand" is used instead of "Expand":
TraditionalForm@Grid@
Table@8
CommutatorExpand@Hx + yLn D
<, 8n, 1, 7, 1<D,
Dividers → AllDD
x+ y
x2 + y2
x2 y + xy2 + x3 + y3
2 x2 y2 + x4 + y4
x4 y + 2 x3 y2 + 2 x2 y3 + xy4 + x5 + y5
3 x4 y2 + 3 x2 y4 + x6 + y6
x6 y + 3 x5 y2 + 3 x4 y3 + 3 x3 y4 + 3 x2 y5 + xy6 + x7 + y7
v7triangles.nb
3
Next all the coefficients are shown, including the zeros, and the threefold structure of the Pauli-Pascal Triangle can be easily
seen:
GridA
TableA9
expanded = CommutatorExpand@Hx + yLn D;
TableACoefficientAexpanded, xn−j ⋅ yj E, 8j, 0, n<E
=, 8n, 1, 9, 1<E,
Dividers → AllE
81, 1<
81, 0, 1<
81, 1, 1, 1<
81, 0, 2, 0, 1<
81, 1, 2, 2, 1, 1<
81, 0, 3, 0, 3, 0, 1<
81, 1, 3, 3, 3, 3, 1, 1<
81, 0, 4, 0, 6, 0, 4, 0, 1<
81, 1, 4, 4, 6, 6, 4, 4, 1, 1<
In the Pauli-Pascal triangle each new number is the addition of the two numbers above it, except when those two numbers
and the number above them are all three equal, in that case the new number becomes zero.
Pascal Triangle for Commuting Operators
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Press [ESC]comm[ESC] in order to show the commutator template.
Here we assign the value of zero to the commutator of these two operators, they commute:
Clear@v, wD;
SetQuantumObject@v, wD;
Pv, wT− = 0
0
Commuting operators reproduce the Pascal triangle, however, the terms do not appear in the standard order. Notice that
"CommutatorExpand" is used instead of "Expand":
4
v7triangles.nb
TraditionalForm@Grid@
Table@8
CommutatorExpand@Hv + wLn D
<, 8n, 1, 7, 1<D,
Dividers → AllDD
v+w
2 vw + v2 + w2
3 v2 w + 3 vw2 + v3 + w3
3
5 v4 w
4 v w + 6 v2 w2 + 4 vw3 + v4 + w4
+ 10 v3 w2 + 10 v2 w3 + 5 vw4 + v5 + w5
6 v5 w + 15 v4 w2 + 20 v3 w3 + 15 v2 w4 + 6 vw5 + v6 + w6
7 v6 w + 21 v5 w2 + 35 v4 w3 + 35 v3 w4 + 21 v2 w5 + 7 vw6 + v7 + w7
Coefficients of the Pascal Triangle for commuting operators
GridA
TableA9
expanded = CommutatorExpand@Hv + wLn D;
TableACoefficientAexpanded, vn−j ⋅ wj E, 8j, 0, n<E
=, 8n, 1, 7, 1<E,
Dividers → AllE
81, 1<
81, 2, 1<
81, 3, 3, 1<
81, 4, 6, 4, 1<
81, 5, 10, 10, 5, 1<
81, 6, 15, 20, 15, 6, 1<
81, 7, 21, 35, 35, 21, 7, 1<
In this Pascal triangle each new number is the addition of the two numbers above it.
Expansions for Operators with a Commutator Equal to One
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Press [ESC]comm[ESC] in order to show the commutator template.
Here we assign the value of one to the commutator of these two operators:
Clear@q, pD;
SetQuantumObject@q, pD;
Pq, pT− = 1
1
v7triangles.nb
Here are the expansions of integer powers of the sum of these two operators. Notice that "CommutatorExpand" is used
instead of "Expand":
TraditionalForm@Grid@
Table@8
CommutatorExpand@Hp + qLn D
<, 8n, 1, 7, 1<D,
Dividers → AllDD
p+q
2 pq + p2 + q2 + 1
3 p2 q + 3 pq2 + p3 + 3 p + q3 + 3 q
4
p3 q
+6
p2 q2
+ 4 pq3 + 12 pq + p4 + 6 p2 + q4 + 6 q2 + 3
5 p4 q + 10 p3 q2 + 10 p2 q3 + 30 p2 q + 5 pq4 + 30 pq2 + p5 + 10 p3 + 15 p + q5 + 10 q3 + 15 q
6 p5 q + 15 p4 q2 + 20 p3 q3 + 60 p3 q + 15 p2 q4 + 90 p2 q2 +
6 pq5 + 60 pq3 + 90 pq + p6 + 15 p4 + 45 p2 + q6 + 15 q4 + 45 q2 + 15
7 p6 q + 21 p5 q2 + 35 p4 q3 + 105 p4 q + 35 p3 q4 + 210 p3 q2 + 21 p2 q5 + 210 p2 q3 +
315 p2 q + 7 pq6 + 105 pq4 + 315 pq2 + p7 + 21 p5 + 105 p3 + 105 p + q7 + 21 q5 + 105 q3 + 105 q
This time the coefficients can be organized in a growing series of Pascal-like triangles:
TraditionalFormAGridA
JoinA9TableAColumnA9"Coefficients of ", pn−2 triangulo−j ⋅ qj =,
Alignment → CenterE, 8triangulo, 0, 4<E=,
TransposeA
TableA
TableA
expanded = CommutatorExpand@Hp + qLn D;
TableA
IfApn−2 triangulo−j ⋅ qj =!= 1,
CoefficientAexpanded, pn−2 triangulo−j ⋅ qj E,
Select@expanded, And@FreeQ@, pD, FreeQ@, qDD &D
E,
8j, 0, n − 2 triangulo<E
, 8n, 1, 9, 1<E, 8triangulo, 0, 4<EEE,
Dividers → All, ItemSize → FullEE
Coefficients of
pn- j q j
Coefficients of
p- j+n-2 q j
Coefficients of
p- j+n-4 q j
81, 1<
8<
8<
81, 2, 1<
81<
8<
81, 3, 3, 1<
83, 3<
8<
81, 4, 6, 4, 1<
86, 12, 6<
83<
81, 5, 10, 10, 5, 1<
810, 30, 30, 10<
815, 15<
81, 6, 15, 20, 15, 6, 1<
815, 60, 90, 60, 15<
845, 90, 45<
81, 7, 21, 35, 35, 21, 7, 1<
821, 105, 210, 210, 105, 21<
8105, 315, 315, 105<
81, 8, 28, 56, 70, 56, 28, 8, 1<
828, 168, 420, 560, 420, 168, 28<
8210, 840, 1260, 840, 210<
81, 9, 36, 84, 126, 126, 84, 36, 9, 1< 836, 252, 756, 1260, 1260, 756, 252, 36< 8378, 1890, 3780, 3780, 1890, 378< 81260
In each Pascal-like triangle each new number is the addition of the two numbers above it times a coefficient that depends on
the triangle and the row.
5
6
v7triangles.nb
Expansions for Operators with an Anticommutator Equal to One
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Press [ESC]anti[ESC] in order to show the anticommutator template (notice the + symbol in the subscript).
Here we assign the value of one to the anticommutator of these two operators:
Clear@r, sD;
SetQuantumObject@r, sD;
Pr, sT+ = 1
1
Here are the expansions of integer powers of the sum of these two operators:
TraditionalForm@Grid@
Table@8
CommutatorExpand@Hr + sLn D
<, 8n, 1, 7, 1<D,
Dividers → AllDD
r+s
r2
2
+ s2 + 1
2
r s + rs + r3 + r + s3 + s
+ r4 + 2 r2 + s4 + 2 s2 + 1
2 r2 s2
r4 s + 2 r3 s2 + 2 r2 s3 + 2 r2 s + rs4 + 2 rs2 + r5 + 2 r3 + r + s5 + 2 s3 + s
3 r4 s2 + 3 r2 s4 + 6 r2 s2 + r6 + 3 r4 + 3 r2 + s6 + 3 s4 + 3 s2 + 1
r6 s
+ 3 r5 s2 + 3 r4 s3 + 3 r4 s + 3 r3 s4 + 6 r3 s2 + 3 r2 s5 +
6 r2 s3 + 3 r2 s + rs6 + 3 rs4 + 3 rs2 + r7 + 3 r5 + 3 r3 + r + s7 + 3 s5 + 3 s3 + s
This time the coefficients can be organized in a growing series of Pauli-Pascal-like triangles:
v7triangles.nb
7
TraditionalFormAGridA
JoinA9TableAColumnA9"Coefficients of ", rn−2 triangulo−j ⋅ sj =,
Alignment → CenterE, 8triangulo, 0, 4<E=,
TransposeA
TableA
TableA
expanded = CommutatorExpand@Hr + sLn D;
TableA
IfArn−2 triangulo−j ⋅ sj =!= 1,
CoefficientAexpanded, rn−2 triangulo−j ⋅ sj E,
Select@expanded, And@FreeQ@, rD, FreeQ@, sDD &D
E,
8j, 0, n − 2 triangulo<E,
8n, 1, 9, 1<E, 8triangulo, 0, 4<EEE,
Dividers → All, ItemSize → FullEE
Coefficients of
Coefficients of
Coefficients of
Coefficients of
rn- j s j
r- j+n-2 s j
r- j+n-4 s j
r- j+n-6 s j
81, 1<
8<
8<
8<
81, 0, 1<
81<
8<
8<
81, 1, 1, 1<
81, 1<
8<
8<
81, 0, 2, 0, 1<
82, 0, 2<
81<
8<
81, 1, 2, 2, 1, 1<
82, 2, 2, 2<
81, 1<
8<
81, 0, 3, 0, 3, 0, 1<
83, 0, 6, 0, 3<
83, 0, 3<
81<
81, 1, 3, 3, 3, 3, 1, 1<
83, 3, 6, 6, 3, 3<
83, 3, 3, 3<
81, 1<
81, 0, 4, 0, 6, 0, 4, 0, 1<
84, 0, 12, 0, 12, 0, 4<
86, 0, 12, 0, 6<
84, 0, 4<
81, 1, 4, 4, 6, 6, 4, 4, 1, 1< 84, 4, 12, 12, 12, 12, 4, 4< 86, 6, 12, 12, 6, 6<
84, 4, 4, 4<
Coefficients of
r- j+n-8 s j
8<
8<
8<
8<
8<
8<
8<
81<
81, 1<
In the Pauli-Pascal triangle each new number is the addition of the two numbers above it times a coefficient that depends on
the triangle and the row, except when those two numbers and the number above them are all three equal, in that case the new
number becomes zero.
Expansions for Operators with a Commutator Equal to Another Operator
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Press [ESC]comm[ESC] in order to show the commutator template.
Here we assign a third operator as the commutator of two operators:
Clear@f, g, hD;
SetQuantumObject@f, g, hD;
Pf, gT− = h
h
8
v7triangles.nb
Expansions evaluate the commutator of f with g to the third operator h, and generate unevaluated commutators of f with h
and g with h.
Please notice the use of Expand and CommutatorExpand in this example:
TraditionalForm@Grid@
Table@8
Expand@CommutatorExpand@Hf + gLn DD
<, 8n, 1, 4, 1<D,
Dividers → AllDD
f +g
2 f g + f 2 + g2 - h
@ f , hD + 2 @g, hD + 3 f 2 g + 3 f g 2 - 3 f h - 3 gh + f 3 + g 3
3 @ f , hDg + g@ f , hD + 5 f @g, hD + @ f , hD f + 3 f @ f , hD + 3 @g, hDg +
5 g@g, hD + 4 f 3 g + 6 f 2 g 2 - 6 f 2 h + 4 f g 3 - 3 f hg - 9 f gh - 6 g 2 h + f 4 + g 4 + 3 h2
Here the three commutators are defined. Notice that defining the commutator Ph, fT− also defines the commutator
Pf, hT−
Please notice the use of Expand and CommutatorExpand in this example:
Clear@f, g, hD;
SetQuantumObject@f, g, hD;
Pf, gT− = h;
Ph, gT− = 0;
Ph, fT− = 0;
TraditionalForm@Grid@
Table@8
Expand@CommutatorExpand@Hf + gLn DD
<, 8n, 1, 4, 1<D,
Dividers → AllDD
f +g
2 f g + f 2 + g2 - h
3 f 2 g + 3 f g 2 - 3 f h - 3 gh + f 3 + g 3
4 f 3 g + 6 f 2 g 2 - 6 f 2 h + 4 f g 3 - 3 f hg - 9 f gh - 6 g 2 h + f 4 + g 4 + 3 h2
Expansions for Operators with Nonnested, Symbolic Commutators
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Here we define c and d as operators without a known commutator and without a known anticommutator
Clear@c, dD;
SetQuantumObject@c, dD;
The expansions are given in terms of the unkwon commutator [c,d].
Notice the use of CommutatorExpand and Expand:
v7triangles.nb
TraditionalForm@Grid@
Table@8
Expand@CommutatorExpand@Hc + dLn DD
<, 8n, 1, 5, 1<D,
Dividers → AllDD
c+d
-@c, dD + 2 cd + c2 + d 2
-@c, dDc - 2 @c, dDd - 2 c@c, dD - d@c, dD + 3 c2 d + 3 cd 2 + c3 + d 3
-@c, dDc2 - 3 c2 @c, dD - 3 @c, dDd 2 - d 2 @c, dD - 2 c@c, dDc - d@c, dDc - 3 @c, dDcd 5 c@c, dDd - 2 d@c, dDd - 3 cd@c, dD + 2 @c, dD2 + 4 c3 d + 6 c2 d 2 + 4 cd 3 + c4 + d 4
-@c, dDc3 - 4 c3 @c, dD - 3 c2 @c, dDc - 2 c@c, dDc2 - d@c, dDc2 - 4 @c, dDc2 d - 9 c2 @c, dDd 6 c2 d@c, dD - 4 @c, dDd 3 - d 3 @c, dD - d 2 @c, dDc - 2 d 2 @c, dDd - 6 @c, dDcd 2 - 9 c@c, dDd 2 - 3 d@c, dDd 2 4 cd 2 @c, dD + 2 @c, dD2 c + 5 @c, dD2 d + 5 c@c, dD2 + 2 d@c, dD2 + 3 @c, dDc@c, dD + 3 @c, dDd@c, dD 3 cd@c, dDc - 7 c@c, dDcd - 3 d@c, dDcd - 7 cd@c, dDd + 5 c4 d + 10 c3 d 2 + 10 c2 d 3 + 5 cd 4 + c5 + d 5
Different (but equivalent) expansions are obtained with the option ReverseOrdering->True inside the CommutatorExpand
command:
TraditionalForm@Grid@
Table@8
Expand@CommutatorExpand@Hc + dLn , ReverseOrdering → TrueDD
<, 8n, 1, 5, 1<D,
Dividers → AllDD
c+d
-@d, cD + 2 dc + c2 + d 2
-2 @d, cDc - @d, cDd - c@d, cD - 2 d@d, cD + 3 dc2 + 3 d 2 c + c3 + d 3
cDc2
-3 @d,
- c2 @d, cD - @d, cDd 2 - 3 d 2 @d, cD - 3 @d, cDdc - 2 c@d, cDc 5 d@d, cDc - c@d, cDd - 2 d@d, cDd - 3 dc@d, cD + 2 @d, cD2 + 4 dc3 + 6 d 2 c2 + 4 d 3 c + c4 + d 4
-4 @d, cDc3 - c3 @d, cD - 2 c2 @d, cDc - 6 @d, cDdc2 - 3 c@d, cDc2 - 9 d@d, cDc2 - c2 @d, cDd 4 dc2 @d, cD - @d, cDd 3 - 4 d 3 @d, cD - 4 @d, cDd 2 c - 9 d 2 @d, cDc - 3 d 2 @d, cDd - c@d, cDd 2 - 2 d@d, cDd 2 6 d 2 c@d, cD + 5 @d, cD2 c + 2 @d, cD2 d + 2 c@d, cD2 + 5 d@d, cD2 + 3 @d, cDc@d, cD + 3 @d, cDd@d, cD 3 c@d, cDdc - 7 d@d, cDdc - 7 dc@d, cDc - 3 dc@d, cDd + 5 dc4 + 10 d 2 c3 + 10 d 3 c2 + 5 d 4 c + c5 + d 5
Expansions for Operators with Nested, Symbolic Commutators
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Here we define i and j as operators without a known commutator and without a known anticommutator
Clear@i, jD;
SetQuantumObject@i, jD;
The option NestedCommutators Ø True inside the command CommutatorExpand specifies that nested commutators can be
generated
9
10
v7triangles.nb
TraditionalForm@Grid@
Table@8
Expand@CommutatorExpand@Hi + jLn , NestedCommutators → TrueDD
<, 8n, 1, 5, 1<D,
Dividers → AllDD
i+ j
-@i, jD + 2 i j + i2 + j2
-3 i@i, jD - 3 j@i, jD + @i, @i, jDD + 2 @ j, @i, jDD + 3 i2 j + 3 i j2 + i3 + j3
-6 i2 @i, jD - 6 j2 @i, jD + 2 @i, @i, jDD j + 4 i@i, @i, jDD + 2 j@i, @i, jDD + 5 i@ j, @i, jDD + 8 j@ j, @i, jDD - 3 i@i, jD j 9 i j@i, jD + 3 @i, jD2 - @i, @i, @i, jDDD - @ j, @i, @i, jDDD - 3 @ j, @ j, @i, jDDD + 4 i3 j + 6 i2 j2 + 4 i j3 + i4 + j4
-10 i3 @i, jD + 10 i2 @i, @i, jDD + 9 i2 @ j, @i, jDD - 11 i2 @i, jD j - 19 i2 j@i, jD - 10 j3 @i, jD + 5 @i, @i, jDD j2 + 3 j2 @i, @i, jDD +
20 j2 @ j, @i, jDD - 7 i@i, jD j2 - 19 i j2 @i, jD + 3 @i, jD2 j - 3 @i, @i, @i, jDDD j - 2 @i, @i, jDD@i, jD - 3 @ j, @i, jDD@i, jD +
15 i@i, jD2 + 11 j@i, jD2 - 8 @i, jD@i, @i, jDD - 5 i@i, @i, @i, jDDD - 2 j@i, @i, @i, jDDD - 10 @i, jD@ j, @i, jDD - 4 i@ j, @i, @i, jDDD 3 j@ j, @i, @i, jDDD - 7 i@ j, @ j, @i, jDDD - 15 j@ j, @ j, @i, jDDD + 11 i@i, @i, jDD j + 2 j@i, @i, jDD j + i@ j, @i, jDD j + @i, jD j@i, jD +
9 i j@i, @i, jDD + 21 i j@ j, @i, jDD - 4 i j@i, jD j + @i, @i, @i, @i, jDDDD + @ j, @i, @i, @i, jDDDD + @ j, @ j, @i, @i, jDDDD +
4 @ j, @ j, @ j, @i, jDDDD + 4 @@i, jD, @i, @i, jDDD + 5 @@i, jD, @ j, @i, jDDD + 5 i4 j + 10 i3 j2 + 10 i2 j3 + 5 i j4 + i5 + j5
Expansions for Operators that Commute with their Commutator
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Here a, b are defined as operators that commute with their commutator:
Clear@a, bD;
SetQuantumObject@a, bD;
Pa, Pa, bT− T− = 0;
Pb, Pa, bT− T− = 0;
The option NestedCommutators Ø True inside the command CommutatorExpand specifies that commutators of commutators
must be generated. In this case, those nested commutators have a value of zero:
TraditionalForm@Grid@
Table@8
Expand@CommutatorExpand@Ha + bLn , NestedCommutators → TrueDD
<, 8n, 1, 5, 1<D,
Dividers → AllDD
a+b
-@a, bD + 2 ab + a2 + b2
-3 a@a, bD - 3 b@a, bD + 3 a2 b + 3 ab2 + a3 + b3
-6 a2 @a, bD - 6 b2 @a, bD - 3 a@a, bDb - 9 ab@a, bD + 3 @a, bD2 + 4 a3 b + 6 a2 b2 + 4 ab3 + a4 + b4
-10 a3 @a, bD - 11 a2 @a, bDb - 19 a2 b@a, bD - 10 b3 @a, bD - 7 a@a, bDb2 - 19 ab2 @a, bD + 3 @a, bD2 b +
15 a@a, bD2 + 11 b@a, bD2 + @a, bDb@a, bD - 4 ab@a, bDb + 5 a4 b + 10 a3 b2 + 10 a2 b3 + 5 ab4 + a5 + b5
v7triangles.nb
11
Expansions for Operators with Nonnested, Symbolic Anticommutators
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Here we define k and m as operators without a known commutator and without a known anticommutator
Clear@k, mD;
SetQuantumObject@k, mD;
The option Anticommutators Ø True inside the command CommutatorExpand specifies that anticommutators should be used
in the expansion
TraditionalForm@Grid@
Table@8
Expand@CommutatorExpand@Hk + mLn , Anticommutators → TrueDD
<, 8n, 1, 5, 1<D,
Dividers → AllDD
k+m
8k, m< + k 2 + m2
8k, m<k + m8k, m< + k 2 m + km2 + k 3 + m3
2
2
2
8k, m<k + k 8k, m< + 8k, m<m + m2 8k, m< + m8k, m<k + 8k, m<km - k8k, m<m + km8k, m< + 2 k 2 m2 + k 4 + m4
8k, m<k 3 + k 2 8k, m<k + m8k, m<k 2 - k 2 8k, m<m + 2 k 2 m8k, m< + m3 8k, m< + m2 8k, m<k +
2 8k, m<km2 - k8k, m<m2 + m8k, m<m2 - 8k, m<2 m - k8k, m<2 + 8k, m<k8k, m< + 8k, m<m8k, m< +
km8k, m<k + k8k, m<km + m8k, m<km + km8k, m<m + k 4 m + 2 k 3 m2 + 2 k 2 m3 + km4 + k 5 + m5
Expansions for Operators with Nested, Symbolic Anticommutators
We load the Quantum Mathematica add-on:
Needs@"Quantum`Notation`"D;
SetQuantumAliases@D;
Here we define t and u as operators without a known commutator and without a known anticommutator
Clear@t, uD;
SetQuantumObject@t, uD;
The options Anticommutators Ø True and NestedCommutators Ø True inside the command CommutatorExpand specifies
that nested anticommutators should be used in the expansion
12
v7triangles.nb
TraditionalForm@Grid@
Table@8
Expand@Expand@
CommutatorExpand@Ht + uLn , Anticommutators → True, NestedCommutators → TrueDDD
<, 8n, 1, 5, 1<D,
Dividers → AllDD
t+u
8t, u< + t2 + u2
-t8t, u< + u8t, u< + 8t, 8t, u<< + t2 u + tu2 + t3 + u3
2 t2 8t, u< + 2 u2 8t, u< - 8t, 8t, u<<u - 2 t8t, 8t, u<< - u8t, 8t, u<< - 4 t8u, 8t, u<< - 2 u8u, 8t, u<< +
2 t8t, u<u + 6 tu8t, u< - 8t, u<2 + 8t, 8t, 8t, u<<< + 2 8u, 8t, 8t, u<<< + 8u, 8u, 8t, u<<< + 2 t2 u2 + t4 + u4
-2 t3 8t, u< + 4 t2 8t, 8t, u<< + 2 t2 8u, 8t, u<< - 4 t2 8t, u<u + 2 t2 u8t, u< + 2 u3 8t, u< - 2 8t, 8t, u<<u2 + 4 u2 8t, 8t, u<< 2 u2 8u, 8t, u<< + 4 t8t, u<u2 - 14 tu2 8t, u< + 2 8t, u<2 u - 2 8t, 8t, 8t, u<<<u - 8u, 8t, 8t, u<<<u - 8t, 8t, u<<8t, u< +
3 8u, 8t, u<<8t, u< - 5 t8t, u<2 - 5 u8t, u<2 - 5 8t, u<8t, 8t, u<< - 3 t8t, 8t, 8t, u<<< - u8t, 8t, 8t, u<<< - 5 8t, u<8u, 8t, u<< 4 t8u, 8t, 8t, u<<< - 8 u8u, 8t, 8t, u<<< - 9 t8u, 8u, 8t, u<<< + u8u, 8u, 8t, u<<< + 5 t8t, 8t, u<<u + 2 t8u, 8t, u<<u +
2 8t, u<u8t, u< + 7 tu8t, 8t, u<< + 22 tu8u, 8t, u<< - 4 tu8t, u<u + 8t, 8t, 8t, 8t, u<<<< + 2 8u, 8t, 8t, 8t, u<<<< +
5 8u, 8u, 8t, 8t, u<<<< + 3 88t, u<, 8t, 8t, u<<< + 88t, u<, 8u, 8t, u<<< + t4 u + 2 t3 u2 + 2 t2 u3 + tu4 + t5 + u5
by José Luis Gómez-Muñoz
http://homepage.cem.itesm.mx/lgomez/quantum/
[email protected]