Posts - MaplePrimes

Quantum Mechanics using Maple's Physics package

by: ecterrab 14455 Product: Maple

December 15 2017

6 0

It passed through my mind it would be interesting to collect the links to the most relevant Mapleprimes posts about Quantum Mechanics using the Physics package of the last couple of years, to have them all accessible from one place. These posts give an idea of what kind of computation is already doable in quantum mechanics, how close is the worksheet input to what we write with paper and pencil, and how close is the typesetting of the output to what we see in textbooks.

At the end of each page linked below, you will see another link to download the corresponding worksheet, that you can open using Maple (say the current version or the version 1 or 2 years ago).

This other set of three consecutive posts develops one problem split into three parts:

This other link is interesting as a quick and compact entry point to the use of the Physics package:

Mini-Course: Computer Algebra for Physicists

There is an equivalent set of Mapleprimes posts illustrating the Physics package tackling problems in General Relativity, collecting them is for one other time.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

The hidden SO(4) symmetry of the Hydrogen Atom

by: ecterrab 14455 Product: Maple

December 14 2017

6 0

December 2017: This is, perhaps, one of the most complicated computations done in this area using the Physics package. To the best of my knowledge never before performed on a computer algebra worksheet. It is exciting to present a computation like this one. At the end the corresponding worksheet is linked so that it can be downloaded and the sections be opened, the computation be reproduced. There is also a link to a pdf with everything open. Special thanks to Pascal Szriftgiser for bringing this problem. To reproduce the computations below, please update the Physics library with the one distributed from the Maplesoft R&D Physics webpage.

June 17, 2020: updated taking advantage of new features of Maple 2020.1 and Physics Updates v.705. Submitted to arxiv.org.

January 25, 2021: updated in the arXiv and submitted for publication in Computer Physics Communications.

Quantum Runge-Lenz Vector and the Hydrogen Atom,

the hidden SO(4) symmetry using Computer Algebra

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab²

(1) University of Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers, Atomes et Molécules, F-59000 Lille, France

(2) Maplesoft

Abstract

Pauli first noticed the hidden SO(4) symmetry for the Hydrogen atom in the early stages of quantum mechanics [1]. Departing from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schrödinger's equation [2]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known [3, 4], its solution involves several steps of manipulating expressions with tensorial quantum operators, simplifying them by taking into account a combination of commutator rules and Einstein's sum rule for repeated indices. Therefore, it is an excellent model to test the current status of CAS concerning this kind of quantum-and-tensor-algebra computations. Generally speaking, when capable, CAS can significantly help with manipulations that, like non-commutative tensor calculus subject to algebra rules, are tedious, time-consuming and error-prone. The presentation also shows a pattern of computer algebra operations that can be useful for systematically tackling more complicated symbolic problems of this kind.

Introduction

The primary purpose of this work is to derive, step-by-step, the SO(4) symmetry of the Hydrogen atom and its spectrum using a computer algebra system (CAS). To the best of our knowledge, such a derivation using symbolic computation has not been shown before. Part of the goal was also to see whether this computation can be performed entering only the main definition formulas, followed by only simplification commands, and without using previous knowledge of the result. The intricacy of this problem is in the symbolic manipulation and simplification of expressions involving noncommutative quantum tensor operators. The simplifications need to take into account commutator rules, symmetries under permutation of indices of tensorial subexpressions, and use Einstein's sum rule for repeated indices.

We performed the derivation using the Maple 2020 system with the Maplesoft Physics Updates v.705. Generally speaking, the default computational domain of CAS doesn't include tensors, noncommutative operators nor related simplifications. On the other hand, the Maple system is distributed with a Physics package that extends that default domain to include those objects and related operations. Physics includes a Simplify command that takes into account custom algebra rules and the sum rule for repeated indices, and uses tensor-simplification algorithms [5] extended to the noncommutative domain.

A note about notation: when working with a CAS, besides the expectation of achieving a correct result for a complicated symbolic calculation, readability is also an issue. It is desired that one be able to enter the definition formulas and computational steps to be performed (the input, preceded by a prompt >, displayed in black) in a way that resembles as closely as possible their paper and pencil representation, and that the results (the output, computed by Maple, displayed in blue) use textbook mathematical-physics notation. The Physics package implements such dedicated typesetting. In what follows, within text and in the output, noncommutative objects are displayed using a different color, e.g. , vectors and tensor indices are displayed the standard way, as in , and , and commutators are displayed with a minus subscript, e.g. . Although the Maple system allows for providing dedicated typesetting also for the input, we preferred to keep visible the Maple input syntax, allowing for comparison with paper and pencil notation. We collected the names of the commands used and a one line description for them in an Appendix at the end. Maple also implements the concept of inert representations of computations, which are activated only when desired. We use this feature in several places. Inert computations are entered by preceding the command with % and are displayed in grey. Finally, as is usual in CAS, every output has an equation label, which we use throughout the presentation to refer to previous intermediate results.

In Sec.1, we recall the standard formulation of the problem and present the computational goal, which is the derivation of the formulas representing the SO(4) symmetry and related spectrum.

In Sec.2, we set tensorial non-commutative operators representing position and linear and angular momentum, respectively , and , their commutation rules used as departure point, and the form of the quantum Hamiltonian . We also derive a few related identities used in the sections that follow.

In Sec.3, we derive the conservation of both angular momentum and the Runge-Lenz quantum operator, respectively and . Taking advantage of the differentialoperators functionality in the Physics package, we perform the derivation exploring two equivalent approaches; first using only a symbolic tensor representation of the momentum operator, then using an explicit differential operator representation for it in configuration space, . With the first approach, expressions are simplified only using the departing commutation rules and Einstein's sum rule for repeated indices. Using the second approach, the problem is additionally transformed into one where the differentiation operators are applied explicitly to a test function . Presenting both approaches is of potential interest as it offers two partly independent methods for performing the same computation, which is helpful to provide confidence on in the results when unknown, a relevant issue when using computer algebra.

In Sec. 4, we derive and show that the classical relation between angular momentum and the Runge-Lenz vectors, = 0, due to the orbital momentum being perpendicular to the elliptic plane of motion while the Runge-Lenz vector lies in that plane, still holds in quantum mechanics, where the components of these quantum vector operators do not commute but =

In Sec. 5, we derive "[Z[a],Z[b]][-]=-(2 i `&hbar;` `ε`[a,b,c] (H L[c]))/`m__e`" using the two alternative approaches described for Sec.3.

In Sec. 6, we derive the well-known formula for the square of the Runge-Lenz vector, Z[k]^2 = 2*H*(`&hbar;`^2+L[a]^2)/m__e+kappa^2 .

Finally, in Sec. 7, we use the SO(4) algebra derived in the previous sections to obtain the spectrum of the Hydrogen atom. Following the literature, this approach is limited to the bound states for which the energy is negative.

Some concluding remarks are presented at the end, and input syntax details are summarized in an Appendix.

1. The hidden SO(4) symmetry of the Hydrogen atom

Let's consider the Hydrogen atom and its Hamiltonian

H = LinearAlgebra[Norm](`#mover(mi("p"),mo("→"))`)^2/(2*m__e)-kappa/r ,

where is the electron momentum, its mass, κ a real positive constant, the distance of the electron from the proton located at the origin, and is its tensorial representation with components [. We assume that the proton's mass is infinite. The electron and nucleus spin are not taken into account. Classically, from the potential , one can derive a central force that drives the electron's motion. Introducing the angular momentum

,

one can further define the Runge-Lenz vector

It is well known that is a constant of the motion, i.e. . Switching to Quantum Mechanics, this condition reads

where, for hermiticity purpose, the expression of must be symmetrized

.

In what follows, departing from the Hamiltonian H, the basic commutation rules between position, momentum and angular momentum in tensor notation, we derive the following commutation rules between the quantum Hamiltonian, angular momentum and Runge-Lenz vector

	=	0
	=	0
	=
	=

Since H commutes with both and , defining

"`M__n`=sqrt(-(`m__e`)/(2 H)) `Z__n`,"

these commutation rules can be rewritten as

	=
	=
	=

This set constitutes the Lie algebra of the SO(4) group.

	2. Setting the problem, commutation rules and useful identities

	3. Commutation rules between the Hamiltonian and each of the angular momentum and Runge-Lenz tensors

	4. Commutation rules between the angular momentum and the Runge-Lenz tensors

	5. Commutation rules between the components of the Runge-Lenz tensor

	6. The square of the norm of the Runge-Lenz vector

	7. The atomic hydrogen spectrum

Conclusions

In this presentation, we derived, step-by-step, the SO(4) symmetry of the Hydrogen atom and its spectrum using the symbolic computer algebra Maple system. The derivation was performed without departing from the results, entering only the main definition formulas in eqs. (1), (2) and (5), followed by using a few simplification commands - mainly Simplify, SortProducts and SubstituteTensor - and a handful of Maple basic commands, subs, lhs, rhs and isolate. The computational path that was used to get the results of sections 2 to 7 is not unique. Instead of searching for the shortest path, we prioritized clarity and illustration of the techniques that can be used to crack problems like this one.

This problem is mainly about simplifying expressions using two different techniques. First, expressions with noncommutative operands in products need reduction with respect to the commutator algebra rules that have been set. Second, products of tensorial operators require simplification using the sum rule for repeated indices and the symmetries of tensorial subexpressions. Those techniques, which are part of the Maple Physics simplifier, together with the SortProducts and SubstituteTensor commands for sorting the operands in products to apply tensorial identities, sufficed. The derivations were performed in a reasonably small number of steps.

Two different computational strategies - with and without differential operators - were used in sections 3 and 5, showing an approach for verifying results, a relevant issue in general when performing complicated algebraic manipulations. The Maple Physics ability to handle differential operators as noncommutative operands in products (as frequently done in paper and pencil computations) facilitates readability and ease in entering the computations. The complexity of those operations is then handled by one Physics:-Library command, ApplyProductsOfDifferentialOperators (see eqs. (47) and (83)).

Besides the Maple Physics ability to handle noncommutative tensor operators and simplify such operators using commutator algebra rules, it is interesting to note: a) the ability of the system to factorize expressions involving products of noncommutative operands (see eqs. (90) and (108)) and b) the extension of the algorithms for simplifying tensorial expressions [5] to the noncommutativity domain, used throughout this presentation.

It is also worth mentioning how equation labels can reduce the whole computation to entering the main definitions, followed by applying a few commands to equation labels. That approach helps to reduce the chance of typographical errors to a very strict minimum. Likewise, the fact that commands and equations distribute over each other allows cumbersome manipulations to be performed in simple ways, as done, for instance, in eqs. (8), (9) and (13).

Finally, it was significantly helpful for us to have the typesetting of results using standard mathematical physics notation, as shown in the presentation above.

Appendix

In this presentation, the input lines are preceded by a prompt > and the commands used are of three kinds: some basic Maple manipulation commands, the main Physics package commands to set things and simplify expressions, and two commands of the Physics:-Library to perform specialized, convenient, operations in expressions.

The basic Maple commands used

•	interface is used once at the beginning to set the letter used to represent the imaginary unit (default is I but we used i).

•	isolate is used in several places to isolate a variable in an expression, for example isolating x in results in

•	lhs and rhs respectively get the left-hand side and right-hand side of an equation

•	subs substitutes the left-hand side of an equation by the righ-hand side in a given target, for example results in

•	@ is used to compose commands. So is the same as . This command is useful to express an abstract combo of manipulations, for example as in (108) ≡ .

The Physics commands used

•	Setup is used to set algebra rules as well as the dimension of space, type of metric, and conventions as the kind of letter used to represent indices.

•	Commutator computes the commutator between two objects using the algebra rules set using Setup. If no rules are known to the system, it outputs a representation for the commutator that the system understands.

•	CompactDisplay is used to avoid redundant display of the functionality of a function.

•	d_[n] represents the tensorial differential operator.

•	Define is used to define tensors, with or without specifying its components.

•	Dagger computes the Hermitian transpose of an expression.

•	Normal, Expand, Factor respectively normalizes, expands and factorizes expressions that involve products of noncommutative operands.

•	Simplify performs simplification of tensorial expressions involving products of noncommutative factors taking into account Einstein's sum rule for repeated indices, symmetries of the indices of tensorial subexpressions and custom commutator algebra rules.

•	SortProducts uses the commutation rules set using Setup to sort the non-commutative operands of a product in an indicated ordering.

The Physics:-Library commands used

•	Library:-ApplyProductsOfDifferentialOperators applies the differential operators found in a product to the product operands that appear to its right. For example, applying this command to results in

•	equalizes the repeated indices in the terms of a sum, so for instance applying this command to results in

References

[1] W. Pauli, "On the hydrogen spectrum from the standpoint of the new quantum mechanics,” Z. Phys. 36, 336–363 (1926)

[2] S. Weinberg, "Lectures on Quantum Mechanics, second edition, Cambridge University Press," 2015.

[3] Veronika Gáliková, Samuel Kováčik, and Peter Prešnajder, "Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space", J. Math. Phys. 54, 122106 (2013)

[4] Castro, P.G., Kullock, R. "Physics of the hydrogen atom". Theor. Math. Phys. 185, 1678–1684 (2015).

[5] L. R. U. Manssur, R. Portugal, and B. F. Svaiter, "Group-Theoretic Approach for Symbolic Tensor Manipulation," International Journal of Modern Physics C, Vol. 13, No. 07, pp. 859-879 (2002).

Download Hidden_SO4_symmetry_of_the_hydrogen_atom.mw

Download Hidden_SO4_symmetry_submitted_to_CPC.pdf (all sections open)

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

lambda(n) argument

by: kenrubenstein 30

December 10 2017

2 0

hi guys....new image that I am very proud of....utilzing the number theoretic argument λ (n)

Bug in dsolve

by: Markiyan Hirnyk 7809 Product: Maple 2017

December 06 2017

0 11

Let us consider

sol1 := dsolve({diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1], 
y(0) = 1}, numeric);
sol1 := proc(x_rkf45) ... end proc

The problem under consideration has the symbolic solution:

sol2 := dsolve({diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1], 
y(0) = 1});

sol2 := y(x) = exp(x*sqrt(2)-x^2)

Let us compare the plots of sol1 and sol2 (which should coincide):

A := plots:-odeplot(sol1, x = 0 .. 1, color = navy, style = point):
B := plot(rhs(sol2), x = 0 .. 1, color = red):
plots:-display([A, B]);

The plots differ after approximately 0.707. Bug_in_dsolve_numeric.mw

Edit. The title and one of the tags.

The Meshless Method

by: Darrell Pepper 50 Product: Maple

December 05 2017

10 7

My co-author and I recently published the 3^rd edition of our finite element book¹ utilizing routines written with MAPLE. In this latest edition, we include a chapter on the meshless method. The meshless method is a unique numerical method for solving PDEs. The finite element method requires the establishment of a mesh associated with node points. Consideration must be given in establishing a good mesh (and minimizing the bandwidth associated with node numbering). The meshless method does not require a mesh to connect nodes. The following excerpt describes the application of the meshless method for a simple 1-D heat transfer simulation using six nodes.

Consider the 1-D expression for heat transfer in a bar defined by the relation²

(1)

where the 1-D domain is bounded by 0 ≤ x ≤ L. The exact solution to this problem is

(2)

with the exact derivative of the temperature given by

(3)

In order to solve the 1-D problem, a multiquadric (MQ) radial basis function (RBF) is used

(4)

where r(x, x_j) is the radial (Euclidean) distance from the expansion point (x_j) to any point (x) , c is a shape parameter that controls the flatness of the RBF and is set by the user, and n is an integer. With n = 1, we retrieve the inverse multiquadric

(5)

that will be used to solve Eq. (1). Other types of RBFs are available; the MQ is accurate and popular.

A global expansion for the 1-D temperature can be expressed as

(6)

with the second derivative of the temperature given as

(7)

Introducing the RBF expansion for the terms in the governing equation, and collocating at the interior points, we obtain

(8)

At the boundaries, we collocate the RBF expansion to impose the boundary conditions

(9)

Defining the operator

(10)

we can now assemble into a fully populated matrix as,

(11)

The solutions obtained using finite difference, finite volume, finite element, boundary element, and the meshless method are listed in Table 1 for 6 equally spaced nodes³ with T_o = 15 and T_L = 25, and L = 1. The interior nodes do not have to be uniformly spaced.

Table 1. Comparison of errors for interior temperatures i = 2,3,…N-1

The Maple code listing follows:

> restart:
   with(LinearAlgebra):with(plots):

# MESHLESS METHOD SOLUTION USING MULTIQUADRIC RADIAL BASIS
FUNCTIONS (RBF) il:=6:T_o:=15:T_L:=25:L:=1:
>   x:=[0,1/5,2/5,3/5,4/5,1]:
>   S:=1000:n:=1:dx:=1/(il-1):
>   C:=Array(1..il,1..il):phi:=Array(1..il,1..il):d2phi:=Array(1..il,1..il):
b:=Vector(1..il):TM:=Vector(1..il):alpha:=Vector(1..il):
for i from 1 to il do
   for j from 1 to il do
      phi[i,j]:=(1+(x[i]-x[j])^2/(S*dx^2))^(n-3/2):
      d2phi[i,j]:=3*((x[j]-x[i])/20)^2/(4*((x[j]-x[i])^2/40+1)^(5/2) )-1/(40*((x[j]-x[i])^2/40+1)^(3/2)):
   end do:
end do:
>   for i from 2 to il-1 do    
>       for j from 1 to il do
>         C[i,j]:=d2phi[i,j]+phi[i,j];
            b[i]:=-x[i];
>         C[1,j]:=phi[1,j];
           C[il,j]:=phi[il,j];
         end do:
      end do:
      b[1]:=T_o:b[il]:=T_L:
> #ConditionNumber(C);
>   alpha:=LinearSolve(convert(C,Matrix),b):
TM[1]:=T_o:TM[6]:=T_L:
for i from 2 to il-1 do
   for j from 1 to il do
>         TM[i]:=TM[i]+alpha[j]*(1+(x[i]-x[j])^2/(S*dx^2))^(n-3/2);
 >     end do:
>   end do:
     evalf(TM);

                     

> TE:=T_o*cos(xx)+(T_L+L-T_o*cos(L))/sin(L)*sin(xx)-xx:
> TE:=subs(TE):
> TE:=plot(TE,xx=0..1,color=blue,legend="Exact",thickness=3):
> MEM:=[seq([subs(x[i]),subs(TM[i])],i=1..6)]:
> T:=plots[pointplot](MEM,style=line,color=red,legend="MEM", thickness=3): 
  MEM:=plots[pointplot](MEM,color=red,legend="MEM",symbol=box, symbolsize=15):

>  plots[display](TE,MEM,T,axes=BOXED,title="Solution - MEM");

Additional examples for two-dimensional domains are described in the text, along with a chapter on the boundary element method. The meshless method is an interesting numerical approach that belongs to the family of weighted residual techniques. The matrix condition number is on the order of 10¹⁰ and can give surprisingly good results – however, the solution can fluctuate when repeatedly executed, eventually returning to the nearly correct solution; this is not an issue when using local assembly instead of the global assemble performed here⁴. The method can be used to form hybrid schemes, e.g., a finite element method can easily be linked with a meshless method to solve a secondary system of equations for problems involving large domains. Results are not sensitive to the location of the nodes; a random placement of points gives qualitatively similar results as a uniform placement.

¹ https://www.maplesoft.com/books/details.aspx?id=615

² https://www.asme.org/products/books/introduction-finite-element-boundary-element

³ ibid

⁴ ibid

Gift Exchange Helper

by: eithne 2072 Product: Maple

December 05 2017

5 2

Just over a year ago, someone asked me if Maple could help them pick names for their family gift exchange, because they were fed up with trying to find a solution by hand that met all their requirements. I knew it could be done, of course, and I spent some time (and at least one family dinner conversation) talking about how to do it. There wasn’t enough time to help my friend last year, but I dusted off my ideas, and my somewhat rusty Maple programming skills, and put something together for this year.

The problem, as stated to me, was “Assign everyone in the group the name of someone else in the group (the person they will buy a present for), with the restriction that no one can be assigned their partner.”

I decided to generalize a bit so that you can specify more than one person in the “do not pick” list for each individual, and the restrictions do not have to be reciprocal. That way, you can use it with rules like “parents cannot pick their children”, or “Elizabeth got Martin two years running, so she can’t pick him again this year”.

Ultimately I went with a “guess and check” approach. For each person, pick a name from the pool of suitable candidates (excluding themselves, anyone on their “do not pick” list, and anyone who has been picked already). Keep assigning names until either everyone has a name, or you end up in a situation where you can’t give someone a name. This can happen, for instance, if Todd is the last name, and the only unmatched name is Catherine, and Todd cannot pick Catherine. If that happens, I tossed all the names back into the virtual hat, gave it a good shake (i.e. randomize()) and tried again. Not as elegant as I would have liked, but it seemed like an effective approach.

It does feel like there ought to be a “nicer” solution. Maybe using graph theory? I know that my code will get into trouble if the restrictions are such that no solution exists. If anyone has any ideas on other/better ways to solve this problem I’d be happy to hear them (now that I’ve had the fun of solving it myself first!).

The application can be found on the application center: Gift Exchange Helper. The name picker algorithm is in the start-up code.

Happy gift giving!

Simplification of products of Dirac matrices

by: ecterrab 14455 Product: Maple

December 04 2017

4 0

The computation of traces of products of Dirac matrices was implemented years ago - see Physics,Trace .

The simplification of products of Dirac matrices, however, was not. Now it is, and illustrating this new feature is the matter of this post. To reproduce the results below please update the Physics library with the one distributed at the Maplesoft R&D Physics webpage.

>

First of all, when loading Physics, a frequent question is about the signature, the default is (- - - +)

>

(1)

This is important because the convention for the Algebra of Dirac Matrices depends on the signature. With the signatures (- - - +) as well as (+ - - -), the sign of the timelike component is 1

>

(2)

With the signatures (+ + + -) as well as (- + + +), the sign of the timelike component is of course -1

>

(3)

The simplification of products of Dirac Matrices, illustrated below with the default signature, works fine with any of these signatures, and works without having to set a representation for the Dirac matrices -- all the results are representation-independent.

The examples below, however, also illustrate a new feature of Physics, for now implemented as a Library:-PerformMatrixOperations command (there is a related, also new, command, , to just present the underlying matrix operations, without performing them). To illustrate this other new functionality , set a representation for the Dirac matrices, say the standard one

>

(4)

The four Dirac matrices are

>

(5)

The definition of the Dirac matrices is implemented in Maple following the conventions of Landau books ([1] Quantum Electrodynamics, V4), and does not depend on the signature, ie the form of these matrices is

>

Array(%id = 18446744078360529726)

(6)

With the default signature, the space part components of change sign when compared with corresponding ones from while the timelike component remains unchanged

>

(7)

>

Array(%id = 18446744078677131982)

(8)

For the default signature, the algebra of the Dirac Matrices, loaded by default when Physics is loaded, is (see page 80 of [1])

>

(9)

When the sign of the timelike component of the signature is -1, we have a -1 factor on the right-hand side of (9).

Note as well that in (9) the right-hand side has no matrix elements. This is standard in particle physics where the computations are performed algebraically, without performing the matrix operations. For the purpose of actually performing the underlying matrix operations, however, one may want to rewrite this algebra including a 4x4 identity matrix. For that purpose, see Algebra of Dirac Matrices with an identity matrix on the right-hand side. For the purpose of this illustration, below we proceed with the algebra as shown in (9), interpreting right-hand sides as if they involve an identity matrix.

Verify the algebra rule by performing all the involved matrix operations

>

(10)

Note that, regarding the spacetime indices, this is a 4x4 matrix, whose elements are in turn 4x4 matrices. Compute first the external 4x4 matrix related to and

>

Matrix(%id = 18446744078587020822)

(11)

Perform now all the matrix operations involved in each of the elements of this 4x4 matrix

>

Matrix(%id = 18446744078743243942)

(12)

By eye everything checks OK.

Consider now the following five products of Dirac matrices

>

(13)

>

(14)

>

(15)

>

(16)

>

(17)

New: the simplification of these products is now implemented

>

(18)

Verify this result performing the underlying matrix operations

>

(19)

>

Matrix(%id = 18446744078553169662) = 4

(20)

The same with the other expressions

>

(21)

>

(22)

>

Array(%id = 18446744078695012102)

(23)

>

Array(%id = 18446744078701714238)

(24)

For e2

>

(25)

>

(26)

>

Matrix(%id = 18446744078470204942)

(27)

>

Matrix(%id = 18446744078550068870)

(28)

For e3 we have

>

(29)

Verify this result,

>

(30)

In this case, with three free spacetime indices lambda, nu, rho, the spacetime components form an array 4x4x4 of 64 components, each of which is a matrix equation

>

Array(%id = 18446744078647326830)

(31)

For instance, the first element is

>

(32)

and it checks OK:

>

Matrix(%id = 18446744078647302614) = Matrix(%id = 18446744078647302974)

(33)

How can you test the 64 components of T all at once?

1. Compute the matrices, without displaying the whole thing, take the elements of the array and remove the indices (ie take the right-hand side); call it M

>

For instance,

>

Matrix(%id = 18446744078629635726) = Matrix(%id = 18446744078629636206)

(34)

Now verify all these matrix equations at once: take the elements of the arrays on each side of the equations and verify that the are the same: we expect for output just

>

(35)

The same for e4

>

(36)

>

(37)

Regarding the spacetime indices this is now an array 4x4x4x4

>

Array(%id = 18446744078730196382)

(38)

For instance the first of these 256 matrix equations

>

(39)

verifies OK:

>

Matrix(%id = 18446744078727227382) = Matrix(%id = 18446744078727227862)

(40)

Now all the 256 matrix equations verified at once as done for e3

>

(41)

Finally, although there is more work to be done here, let's define some tensors and contract their product with these expressions involving products of Dirac matrices.

For example,

>

${A[nu], B[lambda], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}$

(42)

Contract with e1 and e2 and simplify

>

(43)

>

(44)

Download DiracMatricesAndPerformMatrixOperation.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Avoiding Scurvy on a Budget

by: Daniel Cheslo 140 Product: Maple

December 04 2017

3 0

Are you a broke university or college student, living day to day, sustaining yourself on a package of ramen noodles every day? Scared that you’re missing necessary nutrients? Well this is the Maple application for you!

This app, refurbished and modified to fit the budget and nutritional needs of a student, based on this older app: https://www.mapleprimes.com/maplesoftblog/34983-The-Diet-Problem, takes a list of more than 20 foods, and based on nutritional and budget constraints that you choose and can change, will give you the cheapest option for foods that fits your needs!

Maybe you’re not a broke student, but rather someone who wants to keep track of calorie intake or macronutrient intake on a daily basis, well then this app still works for you!

Find it here:
https://www.maplesoft.com/applications/view.aspx?SID=154368

Generator of exercises with vectors

by: Lenin Araujo Castillo 1790 Product: Maple 2017

December 03 2017

3 0

Application to generate vector exercises using very easy to use patterns. These exercises are for sum of vectors in calculations of magnitude, direction, graph and projections. Basically to evaluate our students on the board and thus not lose the results. For university professors and engineering students.

Generator_of_exercises_with_vectors.mw

Lenin Araujo Castillo

Ambassador of Maple

Tagging users - new feature on MaplePrimes

by: Bryon 1728 Product: MaplePrimes

December 01 2017

3 1

We added a small new feature to MaplePrimes that will make it easier to tag other members in your questions, posts and comments.

To use the feature, type "@" in a message. If you wait a moment, a list of names will be populated, starting with members who are participating in the current thread, followed by other members ordered by reputation. You can also begin to type after the "@" to search for a particular member. Similar to the above, members who match your search who are in the current thread will appear first, followed by others.

If you are tagged in this way, an indication will appear in your notifications pane.

We hope you find this new feature useful!

Meet Your Developers - John May

by: RochelleAngyal 270 Product: Maple

November 28 2017

8 3

Many of you enjoyed our profile on one of our developers, Paulina Chin, so we’re happy to bring you another one!

Today, we’ll be talking with John May, Senior Developer of Maple. Let’s get started.

What do you do at Maplesoft?
Until recently I was consulting on-site at the NASA Jet Propulsion Laboratory helping people there more effectively solve their engineering problems using Maplesoft products. But my main job that I am back to full time now is the development and maintenance of various parts of the Maple library.
What did you study in school?
I studied both Pure and Applied Mathematics at the University of Oregon, focusing a lot on Abstract Algebra. In graduate school, I specialized more in computation mathematics like computer algebra and numerical analysis. My Ph.D. work focused on effective numerical algorithms for problems in polynomial algebra – with implementations in Maple!
What area(s) of Maple are you currently focusing on in your development?
Right now I am focused on addressing complaints I’ve gotten from engineers about the usability of units with other parts of the math library.
What’s the coolest feature of Maple that you’ve had a hand in developing?
A lot of the cool things I’ve built live pretty deep in the internals of Maple. I’ve done a lot of meta-heuristic tuning to seamlessly integrate high-performance libraries into top-level Maple commands.

I had a lot of fun developing a lot of the stuff for manipulation and visualization of colors in the ColorTools package.
What do you like most about working at Maplesoft? How long have you worked here?
I started working at Maple in 2007, but I’ve been a Maple user since 1997. I love being part of the magic that brings powerful algorithmic mathematics to everyone. The R&D team is also full of eccentric nerds who are great fun to work with.
Favourite hobby?
It varies by the season, but right now it is prime for mountain biking in southern California. I ride my local trails a couple times a week, and when I get I chance, I love to get away on epic bikepacking adventures (like this one: https://www.bikemag.com/features/two-wheeled-escape-one-hour-from-l-a/ this is me: https://cdn.bikemag.com/uploads/2016/05/16File.jpg ).
What do you like on your pizza?
Anything and everything. Something different every time. My all-time favorite pie my from grad school days is the “Rio Rancho” from the dearly departed That’s Amore Pizza (which was next to the comic book store and across the street from North Carolina State University). It was an olive oil and mozzarella pizza with chopped bacon that was covered in sliced fresh roma tomatoes and drizzled with ranch dressing when it came out of the oven.
What’s your favourite movie?
It’s really hard to pick just one. So, I’ll go with the safe answer and say the greatest movie of all time, and “Weird Al” Yankovic’s only foray into movies, UHF, is my favorite.
http://www.imdb.com/title/tt0098546/
What skill would you love to learn? (That you haven’t already) Why?
Another hard one. I feel like I’ve dabbled in lots of things that I would like to get better at. At the top of the list is probably unicycling. I’d love to get good enough to play Unicyle Football or do Muni (mountain unicyling).
https://en.wikipedia.org/wiki/Mountain_unicycling
http://www.unicyclefootball.com/
Who’s your favourite mathematician?
Batman. https://youtu.be/AcMEckOyoaM

Educational Outreach Champion: Empowering Students...

by: Jennifer Iorgulescu 95 Product: Maple

November 27 2017

2 1

The Perimeter Institute for Theoretical Physics (PI) is a place where bold ideas flourish. It brings together great minds in a shared effort to achieve scientific breakthroughs that will transform our future. PI is an independent research center in foundational theoretical physics.

One of the key mission objectives of Perimeter is to provide scientific training and educational outreach activities to the general public. Maplesoft is proud to be part of this endeavor, as PI’s Educational Outreach Champion. Maplesoft’s contributions support Perimeter’s Teacher Network training activities, core educational resources, development of online course material, support of events such as the EinsteinPlus workshop for high school teachers, the International Summer School for Young Physicists (ISSYP), and other initiatives.

The annual International Summer School for Young Physicists (ISSYP) is a two-week camp that brings together 40 exceptional physics-minded students from high schools across the globe to PI. The program covers many different topics in physics such as quantum mechanics, special relativity, general relativity, and cosmology. Each year students receive a complimentary copy of Maple, and use the product to practice and strengthen their math skills. The program receives an average of three-hundred applications from students in grades eleven and twelve from around the world. Competition is intense, and students who are chosen for the program are extremely bright and advanced for their age; however there is some variation in their level of math and physics knowledge. . Students are asked to review a “math primer” document to prepare them with the background needed for the program.The ISSYP program now uses Möbius, Maplesoft’s online courseware platform to administer this primer. With Möbius, PI has moved from a pdf document primer to fully online material, which has motivated more students to complete the material and be more engaged in their courses. The interactivity and engagement that technology provides has made the summer program more productive and dynamic.

EinsteinPlus is a one-week intensive workshop for Canadian and international high school teachers that focuses on modern physics, including quantum physics, special relativity, and cosmology. EinsteinPlus also provides unique opportunities to learn some of the latest developments in physics from expert researchers at the forefront of their fields. Maplesoft proudly supports this workshop by giving teachers access to and training in Maple.

Perimeter Institute also organizes a lively program of seminars, regularly exposing researchers and students to current ideas in the wider theoretical physics community. The talks provide content outside of, but related to, core disciplines. Recently Maplesoft’s own physics expert Dr. Edgardo Cheb-Terrab conducted a lecture and training session at PI on Computer Algebra for Theoretical Physics.

Dr. Edgardo Cheb-Terrab is the force behind the algorithms and Maple libraries of the ODE and PDE symbolic solvers, the MathematicalFunctions package (an expert system in special functions) and the Physics package, among other things.

In his talk at PI, the Physics project at Maplesoft was presented and the resulting Physics package was illustrated through simple problems in classical field theory, quantum mechanics and general relativity, and through tackling the computations of some recent Physical Review papers in those areas. In addition there was a hands-on workshop where attendees were offered four choices of activity: follow the mini-course; explore items of the worksheet of the morning presentation; or bring their own problems so that Dr. Cheb-Terrab could guide them on how to tackle it using the Physics package and Maple in general.

As a company that strives to continuously improve student learning, and empower instructors and researchers with the tools necessary to compete in an ever changing and demanding educational environment, Maplesoft’s partnership with the Perimeter Institute allows us to do just that. We take great pride and joy in bringing our technology to outreach programs for students and teachers, making these opportunities a more productive and dynamic experience for all.

Algebra of Dirac matrices with an identity matrix...

by: ecterrab 14455 Product: Maple

November 22 2017

4 4

In a previous Mapleprimes question related to Dirac Matrices, I was asked how to represent the algebra of Dirac matrices with an identity matrix on the right-hand side of . Since this is a hot-topic in general, in that, making it work, involves easy and useful functionality however somewhat hidden, not known in general, it passed through my mind that this may be of interest in general. (To reproduce the computations below you need to update your Physics library with the one distributed at the Maplesoft R&D Physics webpage.)

>

First of all, this shows the default algebra rules loaded when you load the Physics package, for the Pauli and Dirac matrices

>

(1)

Now, you can always overwrite these algebra rules.

For instance, to represent the algebra of Dirac matrices with an identity matrix on the right-hand side, one can proceed as follows.

First create the identity matrix. To emulate what we do with paper and pencil, where we write I to represent an identity matrix without having to see the actual table 2x2 with the number 1 in the diagonal and a bunch of 0, I will use the old matrix command, not the new Matrix (see more comments on this at the end). One way of entering this identity matrix is

>

array( 1 .. 4, 1 .. 4, [( 4, 1 ) = (0), ( 1, 2 ) = (0), ( 2, 3 ) = (0), ( 1, 3 ) = (0), ( 2, 2 ) = (1), ( 4, 2 ) = (0), ( 3, 4 ) = (0), ( 1, 4 ) = (0), ( 3, 1 ) = (0), ( 4, 4 ) = (1), ( 3, 2 ) = (0), ( 1, 1 ) = (1), ( 2, 1 ) = (0), ( 4, 3 ) = (0), ( 3, 3 ) = (1), ( 2, 4 ) = (0) ] )

(2)

The most important advantage of the old matrix command is that I is of type algebraic and, consequently, this is the important thing, one can operate with it algebraically and its contents are not displayed:

>

(3)

>

(4)

And so, most commands of the Maple library, that only work with objects of type algebraic, will handle the task. The contents are displayed only on demand, for instance using eval

>

array( 1 .. 4, 1 .. 4, [( 4, 1 ) = (0), ( 1, 2 ) = (0), ( 2, 3 ) = (0), ( 1, 3 ) = (0), ( 2, 2 ) = (1), ( 4, 2 ) = (0), ( 3, 4 ) = (0), ( 1, 4 ) = (0), ( 3, 1 ) = (0), ( 4, 4 ) = (1), ( 3, 2 ) = (0), ( 1, 1 ) = (1), ( 2, 1 ) = (0), ( 4, 3 ) = (0), ( 3, 3 ) = (1), ( 2, 4 ) = (0) ] )

(5)

Returning to the topic at hands: set now the algebra the way you want, with an I matrix on the right-hand side, and without seeing a bunch of 0 and 1

>

(6)

>

$[algebrarules = {%AntiCommutator(Physics:-Dgamma[mu], Physics:-Dgamma[nu]) = 2*Physics:-g_[mu, nu]*`&Iopf;`}]$

(7)

And that is all.

Check it out

>

(8)

Set now a Dirac spinor; this is how you could do that, step-by-step.

Again you can use {vector, matrix, array} or {Vector, Matrix, Array}, and again, if you use the Upper case commands, you always have the components visible, and cannot compute with the object normally since they are not of type algebraic. So I use matrix, not Matrix, and matrix instead of vector so that the Dirac spinor that is both algebraic and matrix, is also displayed in the usual display as a "column vector"

>

$[anticommutativeprefix = {_lambda, psi, :-Psi}]$

(9)

In addition, following your question, in this example I explicitly specify the components of the spinor, in any preferred way, for example here I use

>

array( 1 .. 4, 1 .. 1, [( 4, 1 ) = (psi[4]), ( 3, 1 ) = (psi[3]), ( 1, 1 ) = (psi[1]), ( 2, 1 ) = (psi[2]) ] )

(10)

Check it out:

>

(11)

>

(12)

Let's see all this working together by multiplying the anticommutator equation by

>

(13)

Suppose now that you want to see the matrix form of this equation

>

(14)

The above has the matricial operations delayed; unleash them

>

Physics:-`.`(%AntiCommutator(Physics:-Dgamma[mu], Physics:-Dgamma[nu]), array( 1 .. 4, 1 .. 1, [( 4, 1 ) = (psi[4]), ( 3, 1 ) = (psi[3]), ( 1, 1 ) = (psi[1]), ( 2, 1 ) = (psi[2]) ] )) = 2*Physics:-g_[mu, nu]*(array( 1 .. 4, 1 .. 1, [( 4, 1 ) = (psi[4]), ( 3, 1 ) = (psi[3]), ( 1, 1 ) = (psi[1]), ( 2, 1 ) = (psi[2]) ] ))

(15)

Or directly perform in one go the matrix operations behind (13)

>

Physics:-`.`(%AntiCommutator(Physics:-Dgamma[mu], Physics:-Dgamma[nu]), array( 1 .. 4, 1 .. 1, [( 4, 1 ) = (psi[4]), ( 3, 1 ) = (psi[3]), ( 1, 1 ) = (psi[1]), ( 2, 1 ) = (psi[2]) ] )) = 2*Physics:-g_[mu, nu]*(array( 1 .. 4, 1 .. 1, [( 4, 1 ) = (psi[4]), ( 3, 1 ) = (psi[3]), ( 1, 1 ) = (psi[1]), ( 2, 1 ) = (psi[2]) ] ))

(16)

REMARK: As shown above, in general, the representation using lowercase commands allows you to use `*` or `.` depending on whether you want to represent the operation or perform the operation. For example this represents the operation, as an exact mimicry of what we do with paper and pencil, both regarding input and output

>

(17)

And this performs the operation

>

(18)

Or to only displaying the operation

>

(19)

And to perform all these matricial operations within an algebraic expression,

>

(20)

>

Download DiracAlgebraWithIdentityMatrix.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

number theoretic based 3D image from Ken

by: kenrubenstein 30 Product: Maple

November 20 2017

2 0

Hey guys...

One of my favorite new images rendered in Maple.....

integrates mod congruency arguments like the following, into an 8 stage animated 3D cylindrical coordinates context::

u*mod(x^(1/3)cos(x),23)

How strong are you compared to a powerlifter?

by: Daniel Cheslo 140 Product: Maple

November 14 2017

2 0

As a powerlifter, I constantly find myself calculating my total between my competition lifts, bench, squat, and deadlift. Following that, I always end up calculating my wilks score. The wilks score is a score used to compare lifters between weight classes (https://en.wikipedia.org/wiki/Wilks_Coefficient ), as comparing someone who weighs 59kg in competition like me, to someone who weighs 120kg+, the other end of the spectrum; obviously the 120kg+ lifter is going to massively out-lift me.

So I decided to program a wilks calculator for quick use, rather than needing to go search for one on the internet. For anyone curious about specific scoring, a score of 300+ is very strong for the average gym goer, and is about normal for a powerlifter. A score of 400+ makes you strong for a powerlifter, putting you in the top 250 powerlifters, while 500+ is the very top, as far as unequipped powerlifting goes, including the top 30. For anyone wondering, my best score at my best meet was 390, although given the progress I’ve made in the gym, should be above 400 by my next meet.

Hope you all enjoy!

Find it here: https://maple.cloud#doc=5687076260413440&key=301A440EFD2C4EDD8480D60B5E3147BF40CA460F842942449C939AB8D2E7D679

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Quantum Mechanics using Maple's Physics package

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lambda(n) argument

Bug in dsolve

The Meshless Method

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Simplification of products of Dirac matrices

Avoiding Scurvy on a Budget

Generator of exercises with vectors

Tagging users - new feature on MaplePrimes

Meet Your Developers - John May

Educational Outreach Champion: Empowering Students...

Algebra of Dirac matrices with an identity matrix...

number theoretic based 3D image from Ken

How strong are you compared to a powerlifter?

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