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Pascal4QM

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Physics...

Pascal4QM 478
April 21 2023
2 0

As Edgardo said, Physics is the best environment to perform non-commutative algebra (and indeed, the only way in Maple). In addition, one can define quantum operator, the most general non commutative object, or one can refine the definition with hermitian or unitary operators. A value might also be assigned to the op and used whenever it is required. Here are a few example.

> 

NULL
> 

NULL
> 

restart; with(Physics); interface(imaginaryunit = I)
> 

Setup(quantumoperators = {A, B}, hermitianoperators = {H}, unitaryoperators = {U})

[hermitianoperators = {H}, quantumoperators = {A, B, H, U}, unitaryoperators = {U}]

 (1)

A and B do not commute by default

> 

A*B-A*B

Physics:-`*`(A, B)-Physics:-`*`(B, A)

 (2)
> 

Library:-Commute(A, B)

false

 (3)

For the inverse, just use

> 

Inverse(A)

Physics:-Inverse(A)

 (4)

It is always possible to assign a value. Note however the lower case "matrix"

> 

A := matrix(2, 2, [[1, a], [a, 0]])

array( 1 .. 2, 1 .. 2, [( 1, 1 ) = (1), ( 1, 2 ) = (a), ( 2, 1 ) = (a), ( 2, 2 ) = (0) ] )

 (5)
> 

A

A

 (6)
> 

B := matrix(2, 2, [[1, b], [b, 0]])

array( 1 .. 2, 1 .. 2, [( 1, 1 ) = (1), ( 1, 2 ) = (b), ( 2, 1 ) = (b), ( 2, 2 ) = (0) ] )

 (7)

Using `*` operator leave the product unevaluated

> 

A*B

Physics:-`*`(A, B)

 (8)
> 

Library:-RewriteInMatrixForm(Physics[`*`](A, B))

Physics:-`.`(array( 1 .. 2, 1 .. 2, [( 1, 1 ) = (1), ( 1, 2 ) = (a), ( 2, 1 ) = (a), ( 2, 2 ) = (0) ] ), array( 1 .. 2, 1 .. 2, [( 1, 1 ) = (1), ( 1, 2 ) = (b), ( 2, 1 ) = (b), ( 2, 2 ) = (0) ] ))

 (9)

Or use the `.` operator

> 

A.B

array( 1 .. 2, 1 .. 2, [( 1, 1 ) = (a*b+1), ( 1, 2 ) = (b), ( 2, 1 ) = (a), ( 2, 2 ) = (a*b) ] )

 (10)

Hermintian and unitary op are known

> 

Dagger(H)

H

 (11)
> 

U*Dagger(U)

Physics:-`*`(U, Physics:-Dagger(U))

 (12)
> 

Simplify(Physics[`*`](U, Physics[Dagger](U)))

1

 (13)
> 

NULL

Download Matrix_op.mw

Physics:-Version(1409)...

Pascal4QM 478
March 17 2023
1 0

Hi,

You could try :

Physics:-Version(1409)

1409 is the last version for M2022.2

convert/exp...

Pascal4QM 478
June 02 2022
2 0

Hi,

Maple result seems correct to me. The point is that the returned sum is a Dirac comb. Writting the result as a sum of exp or a sum of Dirac is a matter of choice depending of what you need. Using convert/exp might help:

> 

restart
> 

with(Physics)
> 

with(inttrans)
> 

interface(imaginaryunit = I)
> 

interface(version)

`Standard Worksheet Interface, Maple 2022.1, Windows 10, May 26 2022 Build ID 1619613`

 (1)
> 

delta__t := `assuming`([proc (t) options operator, arrow; Sum(Dirac(t-k*T__0), k = -infinity .. infinity) end proc], [T__0 > 0])

proc (t) options operator, arrow; Sum(Dirac(t-Physics:-`*`(k, T__0)), k = -infinity .. infinity) end proc

 (2)
> 

fourier(delta__t(t), t, omega)

Sum(exp(-I*T__0*k*omega), k = -infinity .. infinity)

 (3)

Considering the Dirac comb

> 

Sum(Dirac(omega-k), k = -infinity .. infinity)

Sum(Dirac(-omega+k), k = -infinity .. infinity)

 (4)

One has the relation

> 

convert(Sum(Dirac(-omega+k), k = -infinity .. infinity), exp) = Sum(Dirac(-omega+k), k = -infinity .. infinity)

Sum(exp((2*I)*Pi*k*omega), k = -infinity .. infinity) = Sum(Dirac(-omega+k), k = -infinity .. infinity)

 (5)

And finally

> 

subs(omega = -omega*T__0/(2*Pi), Sum(exp((2*I)*Pi*k*omega), k = -infinity .. infinity) = Sum(Dirac(-omega+k), k = -infinity .. infinity))

Sum(exp(-I*T__0*k*omega), k = -infinity .. infinity) = Sum(Dirac((1/2)*omega*T__0/Pi+k), k = -infinity .. infinity)

 (6)

Note that the reciprocal conversion is not (yet) supported

> 

Sum(exp((2*I)*Pi*k*omega), k = -infinity .. infinity)

Sum(exp((2*I)*Pi*k*omega), k = -infinity .. infinity)

 (7)
> 

convert(Sum(exp((2*I)*Pi*k*omega), k = -infinity .. infinity), Dirac)

Error, unrecognized conversion: Dirac

  
> 

NULL

Download DiracComb.mw

Possibly Physics:-Fundiff...

Pascal4QM 478
January 09 2022
4 4

Hi,

For deriving an equation in Physics, Fundiff is often the good approach. This should give the result:

> 

restart
> 

with(Physics)
> 

CompactDisplay(Y(t))

Y(t)*`will now be displayed as`*Y

 (1)
> 

ex := (1/2)*m*((diff(Y(t), t))^2+(diff(w(Y(t), t), t))^2)

(1/2)*m*((diff(Y(t), t))^2+((D[1](w))(Y(t), t)*(diff(Y(t), t))+(D[2](w))(Y(t), t))^2)

 (2)
> 

RR := Intc(ex, t)

Int((1/2)*m*((diff(Y(t), t))^2+((D[1](w))(Y(t), t)*(diff(Y(t), t))+(D[2](w))(Y(t), t))^2), t = -infinity .. infinity)

 (3)
> 

Fundiff(RR, Y)

(1/2)*m*(-2*(diff(diff(Y(t), t), t))*(D[1](w))(Y(t), t)^2+(-2*(diff(Y(t), t))^2*(D[1, 1](w))(Y(t), t)-4*(D[1, 2](w))(Y(t), t)*(diff(Y(t), t))-2*(D[2, 2](w))(Y(t), t))*(D[1](w))(Y(t), t)-2*(diff(diff(Y(t), t), t)))

 (4)
> 

NULL

Download FundiffTest.mw

Define quantum operators and use Commuta...

Pascal4QM 478
December 30 2021
2 0

Hi,

In order to have `*` redefined, you need first to define quantum operators for instance. Then [a,b] has the same meaning whether Physics is loaded or not. I guess you'd like to use the commutator. For that purpose, just use "Commutator":

> 

restart
> 

with(Physics)
> 

Setup(quantumoperators = {A, B})

[quantumoperators = {A, B}]

 (1)
> 

R := Commutator(A, B)

Physics:-Commutator(A, B)

 (2)
> 

Expand(R)

Physics:-`*`(A, B)-Physics:-`*`(B, A)

 (3)
> 

NULL

Download CommutatorTest.mw

anticommutativeprefix = {x, y}...

Pascal4QM 478
August 20 2021
2 3

Hi,

In your case, you should rather use Setup(anticommutativeprefix = {x, y}), not "noncommutativeprefix". Then, it is unnecessary to specify "algebrarules={%AntiCommutator(x,x)=0, %AntiCommutator(y,y)=0}" as it will be automatically the case.

Finally, Maple's result:

AntiCommutator(x,y)=0

seems correct to me. As x and y anticommute, x*y + y*x = x*y - x*y = 0.

Physics differentialoperators...

Pascal4QM 478
April 07 2021
2 1

Hi,

A possible way is to define a formal differential operator within the Physics package. It can be define with an arbitrary function and trigged at will. For an exemple, have look at:

https://www.mapleprimes.com/posts/208710-Quantum-Commutation-Rules-Basics

Of course, it must be adapted to your need.

Component...

Pascal4QM 478
January 29 2021
2 0

Hi,

Using "Component", it is possible to reconstruct a Vector:

V := Vector([seq(Component(q_, k), k = 1 .. 3)])

 

Component2Vector.mw

 

Parentheses...

Pascal4QM 478
September 21 2020
0 2

Try -3 - (-5) or  4*(-10), It should work as expected.

 However, I agree that an expression like -3 - -5 should work without parentheses in Maple, mathematically the expression is correct, surprising. Entering -3-+-5 or 4*-10 in Matlab gives the expected result.

Alias and Physics:-Latex...

Pascal4QM 478
September 13 2020
3 4

Hi,

I think that using the new Physics:-Latex with alias, there is even no need for replacing.
 

> 

restart
> 

sol := dsolve(diff(x(t), `$`(t, 3)) = x(t))

x(t) = _C1*exp(t)+_C2*exp(-(1/2)*t)*sin((1/2)*3^(1/2)*t)+_C3*exp(-(1/2)*t)*cos((1/2)*3^(1/2)*t)

 (1)
> 

alias(seq(c[k] = _C || k, k = 1 .. 3))

c[1], c[2], c[3]

 (2)
> 

latex(x(t) = _C1*exp(t)+_C2*exp(-(1/2)*t)*sin((1/2)*3^(1/2)*t)+_C3*exp(-(1/2)*t)*cos((1/2)*3^(1/2)*t))

x \left( t \right) ={\it \_C1}\,{{\rm e}^{t}}+{\it \_C2}\,{{\rm e}^{-{
\frac {t}{2}}}}\sin \left( {\frac {\sqrt {3}t}{2}} \right) +{\it \_C3}
\,{{\rm e}^{-{\frac {t}{2}}}}\cos \left( {\frac {\sqrt {3}t}{2}}
 \right)

 
> 

Physics:-Latex(x(t) = _C1*exp(t)+_C2*exp(-(1/2)*t)*sin((1/2)*3^(1/2)*t)+_C3*exp(-(1/2)*t)*cos((1/2)*3^(1/2)*t))

x \left(t \right) =
c_{1} {\rm e}^{t}+c_{2} {\rm e}^{-\frac{t}{2}} \sin \left(\frac{\sqrt{3} t}{2}\right)+c_{3} {\rm e}^{-\frac{t}{2}} \cos \left(\frac{\sqrt{3} t}{2}\right)

  
> 

``


 

Download PhysicsLatexTest.mw

 

add...

Pascal4QM 478
July 09 2020
1 0

Hi,

use add instead of sum, it should work.

Differential operator...

Pascal4QM 478
June 19 2020
3 1

You might define a differential operator
 

 

> 

restart; with(Physics); with(Physics[Vectors]); interface(imaginaryunit = I)
> 

Setup(differentialoperators = {[N_, [x, y, z]]})

[differentialoperators = {[N_, [x, y, z]]}]

 (1)
> 

CompactDisplay(f(x, y, z))

` f`(x, y, z)*`will now be displayed as`*f

 (2)
> 

N_ := proc (f) options operator, arrow; %Nabla(f) end proc

proc (f) options operator, arrow; %Nabla(f) end proc

 (3)
> 

N_^4*f(x, y, z)

Physics:-`*`(Physics:-`^`(N_, 4), f(x, y, z))

 (4)
> 

Library:-ApplyProductsOfDifferentialOperators(Physics[`*`](Physics[`^`](N_, 4), f(x, y, z)))

%Nabla(%Nabla(%Nabla(%Nabla(f(x, y, z)))))

 (5)
> 

value(%Nabla(%Nabla(%Nabla(%Nabla(f(x, y, z))))))

diff(diff(diff(diff(f(x, y, z), x), x), x), x)+2*(diff(diff(diff(diff(f(x, y, z), x), x), y), y))+2*(diff(diff(diff(diff(f(x, y, z), x), x), z), z))+diff(diff(diff(diff(f(x, y, z), y), y), y), y)+2*(diff(diff(diff(diff(f(x, y, z), y), y), z), z))+diff(diff(diff(diff(f(x, y, z), z), z), z), z)

 (6)
> 

NULL
> 

``


 

Download DiffOp.mw

Setup(quantumoperators = V)...

Pascal4QM 478
June 03 2020
0 0

Hi,

With Physics package, to get this feature, V should be declared as a quantum operator. Also, use Dagger instead of Transpose, or V_^*.

restart;
with(Physics);
with(Physics[Vectors]);
Setup(quantumoperators = V);
V_*Dagger(V_);

This question is related:

https://www.mapleprimes.com/questions/229481-Noncommutative-Product-Of-Matrices#answer268293

 

Add...

Pascal4QM 478
May 30 2020
1 1

Hi,

L := [1, 2, 3, 4, 5];
                      L := [1, 2, 3, 4, 5]

add(L);
                               15

 

Restart...

Pascal4QM 478
May 20 2020
1 0

Hi,

I have the same sometime. Just enter "restart" and this should load the last version, even if the message is the same. Otherwise, just close your worksheet and reopen it, it should work fine.

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