## Bilnov Yevgeniy, 7th grade

by: Maple V

The rocket flies

Быльнов_raketa_letit.mws

## Ibragimova Evelina, 9th grade

by:

Plotting the function of a complex variable

Plotting_the_function_of_a_complex_variable.mws

## Ibragimova Evelina, 9th grade

by: Maple V

Animated 3-D cascade of dolls

3d_matryoshkas_en.mws

## Pearson correlation

Maple 2018

With this application developed entirely in Maple using native syntax and embedded components for science and engineering students. Just replace your data and you're done.

Pearson_Coeficient.mw

Lenin Araujo Castillo

Ambassador of Maple

## Foucault’s Pendulum Exploration Using MAPLE18

by: Maple 18

Foucault’s Pendulum Exploration Using MAPLE18

https://www.ias.ac.in/describe/article/reso/024/06/0653-0659

In this article, we develop the traditional differential equation for Foucault’s pendulum from physical situation and solve it from
standard form. The sublimation of boundary condition eliminates the constants and choice of the local parameters (latitude, pendulum specifications) offers an equation that can be used for a plot followed by animation using MAPLE. The fundamental conceptual components involved in preparing differential equation viz; (i) rotating coordinate system, (ii) rotation of the plane of oscillation and its dependence on the latitude, (iii) effective gravity with latitude, etc., are discussed in detail. The accurate calculations offer quantities up to the sixth decimal point which are used for plotting and animation. This study offers a hands-on experience. Present article offers a know-how to devise a Foucault’s pendulum just by plugging in the latitude of reader’s choice. Students can develop a miniature working model/project of the pendulum.

## Vector Force in space 3d

Maple 2018

Exercises solved online with Maple exclusively in space. I attach the explanation links on my YouTube channel.

Part # 01

https://www.youtube.com/watch?v=8Aa2xzU8LwQ

Part # 02

https://www.youtube.com/watch?v=qyGT28CeSz4

Part # 03

https://www.youtube.com/watch?v=yf8rjSPbv5g

Part # 04

https://www.youtube.com/watch?v=FwHPW7ncZTg

Part # 05

https://www.youtube.com/watch?v=bm3frpukb0I

Link for download the file:

Vector_Exercises-Force_in_space.mw

Lenin AC

Ambassador of Maple

## Maple Conference – New Deadline for Submissions

by: Maple Maple Toolboxes

I just wanted to let everyone know that the Call for Papers and Extended Abstracts deadline for the Maple Conference has been extended to June 14.

The papers and extended abstracts presented at the 2019 Maple Conference will be published in the Communications in Computer and Information Science Series from Springer. We welcome topics that fall into the following broad categories:

• Maple in Education
• Algorithms and Software
• Applications of Maple

You can learn more about the conference or submit your paper or abstract here:

https://www.maplesoft.com/mapleconference/Papers-and-Presentations.aspx

Hope to hear from you soon!

## EXPERIENCE WITH MAPLESIM

MapleSim 2018

It is a very good computational tool to perform modeling and simulation using our world as a reference. You can also teach math knowing how to choose the right icons.
I recommend this software to everyone who wants to simulate objects or multibodies. In any case, knowledge of physics and mathematics, especially vector mechanics, is necessary.
Very grateful to the Maplesoft company for sharing their projects through the MapleSim gallery.

From now on all projects will be with Maple and MapleSim.

Lenin AC

Ambassador of Maple

## Maple Conference - Call for Papers and Extended...

by: Maple

Submit your paper or extended abstract to the Maple Conference!

The papers and extended abstracts presented at the 2019 Maple Conference will be published in the Communications in Computer and Information Science Series from Springer.

The deadline to submit is May 27, 2019.

This conference is an amazing opportunity to contribute to the development of technology in academics. I hope that you, or your colleagues and associates, will consider making a contribution.

We welcome topics that fall into the following broad categories:

• Maple in Education
• Algorithms and Software
• Applications of Maple

You can learn more about the conference or submit your paper or abstract here:

https://www.maplesoft.com/mapleconference/Papers-and-Presentations.aspx

## Introducing the Maple Quantum Chemistry Toolbox

Quantum Chemistry Toolbox from RDMChem

Maple 2019 has a new add-on package Maple Quantum Chemistry Toolbox from RDMChem for computing the energies and properties of molecules.  As a member of the team at RDMChem that developed the package, I would like to tell the story of its origins and provide a brief demonstration of the package.

Thinking about Quantum Chemistry at Harvard

The story of the Maple Quantum Chemistry Toolbox begins with my graduate studies in Chemical Physics at Harvard University in the late 1990s.  Even in 1998 programs for computing the energies and properties of molecules were extremely complicated and nonintuitive.  Many of the existing programs had begun in the 1970s on computers whose programs would be recorded on punchcards.

Fig. 1: Used Punchcard by Pete Birkinshaw from Manchester, UK CC BY 2.0

Even today some of these programs have remnants of their early versions such as input files that must start on the second column to account for the margin of the now non-existent punchcards.  As a student, I made a bound copy of one of these manuals at a local Kinkos photocopy shop and later found myself in Harvard Yard, thinking that there must be a better way to present quantum chemistry computations.  The idea for a Maple-like package for quantum chemistry was born in that moment.

At the same time I was learning about something called the two-electron reduced density matrix (2-RDM).  The basic variable in quantum chemistry is the wave function which is the probability amplitude for finding each of the electrons in a molecule.  Because electrons are indistinguishable with pairwise interactions, the wave function contains much more information than is needed for computing the energies and electronic properties of molecules.  The energies and properties of any molecule with any number of electrons can be expressed as a function of a 2 electron matrix, the 2-RDM [1-3].  A quantum chemistry based on the 2-RDM, it was known, would have potentially significant advantages over wave function calculations in terms of accuracy and computational cost, especially for molecules far from the mean-field limit.  A 2-RDM approach to quantum chemistry became the focus of my Ph.D. thesis.

Representing Many Electrons with Only Two Electrons

The idea of using the 2-RDM in quantum chemistry can be attributed to four scientists: two physicists Kodi Husimi and Joseph Mayer, a chemist Per-Olov Lowdin, and a mathematician John Coleman [1-3].  In the early 1940s Husimi first published the idea in a Japanese physics journal, but in the midst of World War II the paper was not widely disseminated in the West.  In the summer of 1951 John Coleman, which attending a physics conference at Chalk River, realized that the ground-state energy of any atom or molecule could be expressed as functional of the 2-RDM, and similar ideas later occurred to Per-Olov Lowdin and Joseph Mayer who published their ideas in Physical Review in 1955.  It was soon recognized that computing the ground-state energy of an atom or molecule with the 2-RDM was potentially difficult because not every two-electron density matrix corresponds to an N-electron density matrix or wave function.  The search for the appropriate constraints on the 2-RDM, known as N-representability conditions, became known as the N-representability problem [1-3].

Beginning in the late 1990s and early 2000s, Carmela Valdemoro and Diego Alcoba at the Consejo Superior de Investigaciones Científicas (Madrid, Spain), Hiroshi Nakatsuji, Koji Yasuda, and Maho Nakata at Kyoto University (Kyoto, Japan), Jerome Percus and Bastiaan Braams at the Courant Institute (New York, USA), John Coleman and Robert Erdahl at Queens University (Kingston, Canada), and my research group and I at The University of Chicago (Chicago, USA) began to make significant progress in the computation of the 2-RDM without computing the many-electron wave function [1-3].  Further contributions were made by Eric Cances and Claude Le Bris at CERMICS, Ecole Nationale des Ponts et Chaussées (Marne-la-Vallée, France), Paul Ayers at McMaster University (Hamilton, Canada), and Dimitri Van Neck at the University of Ghent (Ghent, Belgium) and their research groups.  By 2014 several powerful 2-RDM methods had emerged for the computation of molecules.  The Army Research Office (ARO) issued a proposal call for a company to develop a modern, built-from-scratch package for quantum chemistry that would contain two newly developed 2-RDM-based methods from our group: the parametric 2-RDM method [1] and the variational 2-RDM method with a fast algorithm for solving the semidefinite program [4,5,6].   The company RDMChem LLC was founded to work with the ARO to develop such a package built around RDMs, and hence, the name of the company RDMChem was selected as a hybrid of the RDM abbreviation for Reduced Density Matrices and the Chem colloquialism for Chemistry.  To achieve a really new design for an electronic structure package with access to numeric and symbolic computations as well as advanced visualizations, the team at RDMChem and I developed a partnership with Maplesoft to build something new that became the Maple Quantum Chemistry Package (or Toolbox), which was released with Maple 2019 on Pi Day.

Maple Quantum Chemistry Toolbox

The Maple Quantum Chemistry Toolbox provides a powerful, parallel platform for quantum chemistry calculations that is directly integrated into the Maple 2019 environment.  It is optimized for both cutting-edge research as well as chemistry education.  The Toolbox can be used from the worksheet, document, or command-line interfaces.  Plus there is a Maplet interface for rapid exploration of molecules and their properties.  Figure 2 shows the Maplet interface being applied to compute the ground-state energy of 1,3-dibromobenzene by density functional theory (DFT) in a 6-31g basis set.

Fig. 2: Maplet interface to the Quantum Chemistry Toolbox 2019, showing a density functional theory (DFT) calculation

After entering a name into the text box labeled Name, the user can click on: (1) the button Web to import the geometry from an online database containing more than 96 million molecules,  (2) the button File to read the geometry from a standard XYZ file, or (3) the button Input to enter the geometry.  As soon the geometry is entered, the Maplet displays a 3D picture of the molecule in the window on the right of the options.  Dropdown menus allow the user to select the basis set, the electronic structure method, and a boolean for geometry optimization.  The user can click on the Compute button to perform the computation.  When the quantum computation completes, the total energy appears in the box labeled Total Energy.  The dropdown menu Analyze contains a list of data tables, plots, and animations that can be selected and then displayed by clicking the Analyze button.  The Maplet interface contains nearly all of the options available in the worksheet interface.   The Help Pages of the Toolbox include extensive curricula and lessons that can be used in undergraduate, graduate, and even high school chemistry courses.  Next we look at some sample calculations in the worksheet interface.

Reproducing an Early 2-RDM Calculation

One of the earliest variational calculations of the 2-RDM was performed in 1975 by Garrod, Mihailović,  and  Rosina [1-3].  They minimized the electronic ground state of the 4-electron atom beryllium as a functional of only two electrons, the 2-RDM.  They imposed semidefinite constraints on the particle-particle (D), hole-hole (Q), and particle-hole (G) metric matrices.  They solved the resulting optimization problem of minimizing the energy as a linear function of the 2-RDM subject to the semidefinite constraints, known as a semidefinite program, by a cutting-plane algorithm.  Due to limitations of the cutting-plane algorithm and computers circa 1975, the calculation was a difficult one, likely taking a significant amount of computer time and memory.

With the Quantum Chemistry Toolbox we can use the command Variational2RDM to reproduce the calculation on a Windows laptop.  First, in a Maple 2019 worksheet we load the commands of the Add-on Quantum Chemistry Toolbox:

 > with(QuantumChemistry);
 (1.1)

Then we define the atom (or molecule) using a Maple list of lists that we assign to the variable atom:

 > atom := [["Be",0,0,0]];
 (1.2)

We can then perform the variational 2-RDM method with the Variational2RDM command to compute the ground-state energy and properties of beryllium in a minimal basis set like the one used by Rosina and his collaborators.  By default the method uses the D, Q, and G N-representability conditions and the minimal "sto-3g" basis set.  The calculation, which completes in seconds, contains a wealth of information in the form of a convenient Maple table that we assign to the variable data.

 > data := Variational2RDM(atom);
 (1.3)

The table contains the total ground-state energy of the beryllium atom in the atomic unit of energy (hartrees)

 > data[e_tot];
 (1.4)

We also have the atomic orbitals (AOs) employed in the calculation

 > data[aolabels];
 (1.5)

as well as the Mulliken populations of these orbitals

 > data[populations];
 (1.6)

We see that 2 electrons are located in the 1s orbital, 1.8 electrons in the 2s orbital, and about 0.2 electrons in the 2p orbitals.  By default the calculation also returns the 1-RDM

 > data[rdm1];
 (1.7)

The eigenvalues of the 1-RDM are the natural orbital occupations

 > LinearAlgebra:-Eigenvalues(data[rdm1]);
 (1.8)

We can display the density of the 2s-like 2nd natural orbital using the DensityPlot3D command providing the atom, the data, and the orbitalindex keyword

 > DensityPlot3D(atom,data,orbitalindex=2);

Similarly,  using the DensityPlot3D command, we can readily display the 2p-like 3rd natural orbital

 > DensityPlot3D(atom,data,orbitalindex=3);

By using Maple keyword arguments in the Variational2RDM command, we can readily change the basis set, use point-group symmetry, add active orbitals with or without self-consistent-field, change the N-representability conditions, as well as explore many other options.  Having reenacted one of the first variational 2-RDM calculations ever, let's examine a more complicated molecule.

Explosive TNT

We consider the molecule TNT that is used as an explosive. Using the command MolecularGeometry, we can import the experimental geometry of TNT from the online PubChem database.

 > mol := MolecularGeometry("TNT");
 (1.9)

The command PlotMolecule generates a 3D ball-and-stick plot of the molecule

 > PlotMolecule(mol);

We perform a variational calculation of the 2-RDM of TNT in an active space of 10 electrons and 10 orbitals by setting the keyword active to the list [10,10].  The keyword casscf is set to true to optimize the active orbitals during the calculation.  The keyword basis is used to set the basis set to a minimal basis set sto-3g for illustration.

 > data := Variational2RDM(mol, active=[10,10], casscf=true, basis="sto-3g");
 (1.10)

The ground-state energy of TNT in hartrees is

 > data[e_tot];
 (1.11)

Unlike beryllium, the electric dipole moment of TNT in debyes is nonzero

 > data[dipole];
 (1.12)

We can easily visualize the dipole moment relative to the molecule's ball-and-stick model with the DipolePlot command

 > DipolePlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

The 1-RDM is returned by default

 > data[rdm1];
 (1.13)

The natural molecular-orbital (MO) occupations are the eigenvalues of the 1-RDM

 > data[mo_occ];
 (1.14)

All of the occupations can be viewed at once by converting the Vector to a list

 > convert(data[mo_occ], list);
 (1.15)

We can visualize these occupations with the MOOccupationsPlot command

 > MOOccupationsPlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

The occupations, we observe, show significant deviations from 0 and 2, indicating that the electrons have substantial correlation beyond the mean-field (Hartree-Fock) limit.  The blue lines indicate the first N/2 spatial orbitals where N is the total number of electrons while the red lines indicate the remaining spatial orbitals.  We can visualize the highest "occupied" molecular orbital (58) with the DensityPlot3D command

 > DensityPlot3D(mol,data, orbitalindex=58);

Similarly, we can visualize the lowest "unoccupied" molecular orbital (59) with the DensityPlot3D command

 > DensityPlot3D(mol,data, orbitalindex=59);
 >

Comparison of orbitals 58 and 59 reveals an increase in the number of nodes (changes in the phase of the orbitals denoted by green and purple), which reflects an increase in the energy of the orbital.

Looking Ahead

The Maple Quantum Chemistry Toolbox 2019, an new Add-on for Maple 2019 from RDMChem, provides a easy-to-use, research-grade environment for the computation of the energies and properties of atoms and molecules.  In this blog we discussed its origins in graduate research at Harvard, its reproduction of an early 2-RDM calculation of beryllium, and its application to the explosive molecule TNT.  We have illustrated only some of the many features and electronic structure methods of the Maple Quantum Chemistry package.  There is much more chemistry and physics to explore.  Enjoy!

Selected References

[1] D. A. Mazziotti, Chem. Rev. 112, 244 (2012). "Two-electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and Physics"

[2]  Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules (Adv. Chem. Phys.) ; D. A. Mazziotti, Ed.; Wiley: New York, 2007; Vol. 134.

[3] A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer-Verlag,  New York, 2000).

[4] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011). "Large-scale Semidefinite Programming for Many-electron Quantum Mechanics"

[5] A. W. Schlimgen, C. W. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 7, 627-631 (2016). "Entangled Electrons Foil Synthesis of Elusive Low-Valent Vanadium Oxo Complex"

[6] J. M. Montgomery and D. A. Mazziotti, J. Phys. Chem. A 122, 4988-4996 (2018). "Strong Electron Correlation in Nitrogenase Cofactor, FeMoco"

Download QCT2019_PrimesV17_05.05.19.mw

## Danish primary schools are using Maple to revoluti...

by: Maple

I recently had a wonderful and valuable opportunity to meet with some primary school students and teachers at Holbaek by Skole in Denmark to discuss the use of technology in the classroom. The Danish education system has long been an advocate of using technology and digital learning solutions to augment learning for its students. One of the technology solutions they are using is Maple, Maplesoft’s comprehensive mathematics software tool designed to meet the unique and complex needs of STEM courses. It is rare to find Maple being used at the primary school level, so it was fascinating to see first-hand how Maple is being incorporated at the school.

In speaking with some of the students, I asked them what their education was like before Maple was incorporated into their course. They told me that before they had access to Maple, the teacher would put an example problem on the whiteboard and they would have to take notes and work through the solution in their notebooks. They definitely prefer the way the course is taught using Maple. They love the fact that they have a tool that let them work through the solution and provide context to the answer, as opposed to just giving them the solution. It forces them to think about how to solve the problem. The students expressed to me that Maple has transformed their learning and they cannot imagine going back to taking lectures using a whiteboard and notebook.

Here, I am speaking with some students about how they have adapted Maple to meet their needs ... and about football. Their team had just won 12-1.

Mathematics courses, and on a broader level, STEM courses, deal with a lot of complex materials and can be incredibly challenging. If we are able to start laying the groundwork for competency and understanding at a younger age, students will be better positioned for not only higher education, but their careers as well. This creates the potential for stronger ideas and greater innovation, which has far-reaching benefits for society as a whole.

Jesper Estrup and Gitte Christiansen, two passionate primary school teachers, were responsible for introducing Maple at Holbaek by Skole. It was a pleasure to meet with them and discuss their vision for improving mathematics education at the school. They wanted to provide their students experience with a technology tool so they would be better equipped to handle learning in the future. With the use of Maple, the students achieved the highest grades in their school system. As a result of this success, Jesper and Gitte decided to develop primary school level content for a learning package to further enhance the way their students learn and understand mathematics, and to benefit other institutions seeking to do the same. Their efforts resulted in the development of Maple-Skole, a new educational tool, based on Maple, that supports mathematics teaching for primary schools in Denmark.

Maplesoft has a long-standing relationship with the Danish education system. Maple is already used in high schools throughout Denmark, supported by the Maple Gym package. This package is an add-on to Maple that contains a number of routines to make working with Maple more convenient within various topics. These routines are made available to students and teachers with a single command that simplifies learning. Maple-Skole is the next step in the country’s vision of utilizing technology tools to enhance learning for its students. And having the opportunity to work with one tool all the way through their schooling will provide even greater benefit to students.

(L-R) Henrik and Carolyn from Maplesoft meeting with Jesper and Gitte from Holbaek by Skole

It helps foster greater knowledge and competency in primary school students by developing a passion for mathematics early on. This is a big step and one that we hope will revolutionize mathematics education in the country. It is exciting to see both the great potential for the Maple-Skole package and the fact that young students are already embracing Maple in such a positive way.

For us at Maplesoft, this exciting new package provides a great opportunity to not only improve upon our relationships with educational institutions in Denmark, but also to be a part of something significant, enhancing the way students learn mathematics. We strongly believe in the benefits of Maple-Skole, which is why it will be offered to schools at no charge until July 2020. I truly believe this new tool has the potential to revolutionize mathematics education at a young age, which will make them better prepared as they move forward in their education.

## System of Comptable-Determined Equations of 2x2...

Maple 2018

This application solves a set of compatible equations of two variables. It also graphs the intersection point of the variable "x" and "y". If we want to observe the intersection point closer we will use the zoom button that is activated when manipulating the graph. If we want to change the variable ("x" and "y") we enter the code of the button that solves and graphs. In spanish.

System_of_Equations_Determined_Compatible_2x2_and_3x3.mw

Lenin Araujo Castillo

Ambassador of Maple

## The Maple Conference is Back!

by: Maple Maple Toolboxes

It is my pleasure to announce the return of the Maple Conference! On October 15-17th, in Waterloo, Ontario, Canada, we will gather a group of Maple enthusiasts, product experts, and customers, to explore and celebrate the different aspects of Maple.

Specifically, this conference will be dedicated to exploring Maple’s impact on education, new symbolic computation algorithms and techniques, and the wide range of Maple applications. Attendees will have the opportunity to learn about the latest research, share experiences, and interact with Maple developers.

In preparation for the conference we are welcoming paper and extended abstract submissions. We are looking for presentations which fall into the broad categories of “Maple in Education”, “Algorithms and Software”, and “Applications of Maple” (a more extensive list of topics can be found here).

You can learn more about the event, plus find our call-for-papers and abstracts, here: https://www.maplesoft.com/mapleconference/

## Overview of the Physics Updates

by: Maple

Overview of the Physics Updates

One of the problems pointed out several times about the Physics package documentation is that the information is scattered. There are the help pages for each Physics command, then there is that page on Physics conventions, one other with Examples in different areas of physics, one "What's new in Physics" page at each release with illustrations only shown there. Then there are a number of Mapleprimes post describing the Physics project and showing how to use the package to tackle different problems. We seldomly find the information we are looking for fast enough.

This post thus organizes and presents all those elusive links in one place. All the hyperlinks below are alive from within a Maple worksheet. A link to this page is also appearing in all the Physics help pages in the future Maple release. Comments on practical ways to improve this presentation of information are welcome.

Description

As part of its commitment to providing the best possible environment for algebraic computations in Physics, Maplesoft launched, during 2014, a Maple Physics: Research and Development website. That enabled users to ask questions, provide feedback and download updated versions of the Physics package, around the clock.

The "Physics Updates" include improvements, fixes, and the latest new developments, in the areas of Physics, Differential Equations and Mathematical Functions. Since Maple 2018, you can install/uninstall the "Physics Updates" directly from the MapleCloud .

Maplesoft incorporated the results of this accelerated exchange with people around the world into the successive versions of Maple. Below there are two sections

 • The Updates of Physics, as  an organized collection of links per Maple release, where you can find a description with examples of the subjects developed in the Physics package, from 2012 till 2019.
 • The Mapleprimes Physics posts, containing the most important posts describing the Physics project and showing the use of the package to tackle problems in General Relativity and Quantum Mechanics.

The update of Physics in Maple 2018 and back to Maple 16 (2012)

 • Physics Updates during 2018
 a.
 b.
 c.
 d. Multivariable Taylor series of expressions involving anticommutative (Grassmannian) variables
 e. New SortProducts command
 f.

 • Physics Maple 2018 updates
 g. Automatic handling of collision of tensor indices in products
 h. User defined algebraic differential operators
 i. The Physics:-Cactus package for Numerical Relativity
 j. Automatic setting of the EnergyMomentumTensor for metrics of the database of solutions to Einstein's equations
 k. Minimize the number of tensor components according to its symmetries, relabel, redefine or count the number of independent tensor components
 l. New functionality and display for inert names and inert tensors
 m. Automatic setting of Dirac, Paul and Gell-Mann algebras
 n. Simplification of products of Dirac matrices
 o. New Physics:-Library commands to perform matrix operations in expressions involving spinors with omitted indices
 p. Miscellaneous improvements

 • Physics Maple 2017 updates
 q. General Relativity: classification of solutions to Einstein's equations and the Tetrads package
 r. The 3D metric and the ThreePlusOne (3 + 1) new Physics subpackage
 s. Tensors in Special and General Relativity
 t. The StandardModel new Physics subpackage

 • Physics Maple 2016 updates
 u. Completion of the Database of Solutions to Einstein's Equations
 v. Operatorial Algebraic Expressions Involving the Differential Operators d_[mu], D_[mu] and Nabla
 w. Factorization of Expressions Involving Noncommutative Operators
 x. Tensors in Special and General Relativity
 y. Vectors Package
 z. New Physics:-Library commands
 aa. Redesigned Functionality and Miscellaneous

 • Physics Maple 2015 updates
 ab. Simplification
 ac. Tensors
 ad. Tetrads in General Relativity
 ae. More Metrics in the Database of Solutions to Einstein's Equations
 af. Commutators, AntiCommutators, and Dirac notation in quantum mechanics
 ag. New Assume command and new enhanced Mode: automaticsimplification
 ah. Vectors Package
 ai. New Physics:-Library commands
 aj. Miscellaneous

 • Physics Maple 18 updates
 ak. Simplification
 al. 4-Vectors, Substituting Tensors
 am. Functional Differentiation
 an. More Metrics in the Database of Solutions to Einstein's Equations
 ao. Commutators, AntiCommutators
 ap. Expand and Combine
 aq. New Enhanced Modes in Physics Setup
 ar. Dagger
 as. Vectors Package
 at. New Physics:-Library commands
 au. Miscellaneous

 • Physics Maple 17 updates
 av. Tensors and Relativity: ExteriorDerivative, Geodesics, KillingVectors, LieDerivative, LieBracket, Antisymmetrize and Symmetrize
 aw. Dirac matrices, commutators, anticommutators, and algebras
 ax. Vector Analysis
 ay. A new Library of programming commands for Physics

 • Physics Maple 16 updates
 az. Tensors in Special and General Relativity: contravariant indices and new commands for all the General Relativity tensors
 ba. New commands for working with expressions involving anticommutative variables and functions: Gtaylor, ToFieldComponents, ToSuperfields
 bb. Vector Analysis: geometrical coordinates with funcional dependency

Mapleprimes Physics posts

 1
 2
 3
 4
 5
 6

 • General Relativity

 7
 8
 9
 10
 11
 12
 13 Minimize the number of tensor components according to its symmetries
 • Quantum Mechanics

 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27

 • Physics package generic functionality

 28
 29
 See Also Conventions used in the Physics package , Physics , Physics examples , A Complete Guide for Tensor computations using Physics

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

## The Zassenhaus formula and the Pauli matrices

by: Maple

The Zassenhaus formula and the algebra of the Pauli matrices

Edgardo S. Cheb-Terrab1 and Bryan C. Sanctuary2

(1) Maplesoft

(2) Department of Chemistry, McGill University, Montreal, Quebec, Canada

 The implementation of the Pauli matrices and their algebra were reviewed during 2018, including the algebraic manipulation of nested commutators, resulting in faster computations using simpler and more flexible input. As it frequently happens, improvements of this type suddenly transform research problems presented in the literature as untractable in practice, into tractable.
 As an illustration, we tackle below the derivation of the coefficients entering the Zassenhaus formula shown in section 4 of [1] for the Pauli matrices up to order 10 (results in the literature go up to order 5). The computation presented can be reused to compute these coefficients up to any desired higher-order (hardware limitations may apply). A number of examples which exploit this formula and its dual, the Baker-Campbell-Hausdorff formula, occur in connection with the Weyl prescription for converting a classical function to a quantum operator (see sec. 5 of [1]), as well as when solving the eigenvalue problem for classes of mathematical-physics partial differential equations [2].   To reproduce the results below - a worksheet with this contents is linked at the end - you need to have your Maple 2018.2.1 updated with the Maplesoft Physics Updates version 280 or higher.

References

 [1] R.M. Wilcox, "Exponential Operators and Parameter Differentiation in Quantum Physics", Journal of Mathematical Physics, V.8, 4, (1967.
 [2] S. Steinberg, "Applications of the lie algebraic formulas of Baker, Campbell, Hausdorff, and Zassenhaus to the calculation of explicit solutions of partial differential equations", Journal of Differential Equations, V.26, 3, 1977.
 [3] K. Huang, "Statistical Mechanics", John Wiley & Sons, Inc. 1963, p217, Eq.(10.60).

Formulation of the problem

The Zassenhaus formula expresses  as an infinite product of exponential operators involving nested commutators of increasing complexity

=

Given ,  and their commutator , if  and  commute with ,  for  and the Zassenhaus formula reduces to the product of the first three exponentials above. The interest here is in the general case, when  and , and the goal is to compute the Zassenhaus coefficients in terms of ,  for arbitrary finite n. Following [1], in that general case, differentiating the Zassenhaus formula with respect to  and multiplying from the right by  one obtains

This is an intricate formula, which however (see eq.(4.20) of [1]) can be represented in abstract form as

from where an equation to be solved for each  is obtained by equating to 0 the coefficient of . In this formula, the repeated commutator bracket is defined inductively in terms of the standard commutator by

and higher-order repeated-commutator brackets are similarly defined. For example, taking the coefficient of  and  and respectively solving each of them for  and  one obtains

This method is used in [3] to treat quantum deviations from the classical limit of the partition function for both a Bose-Einstein and Fermi-Dirac gas. The complexity of the computation of  grows rapidly and in the literature only the coefficients up to  have been published. Taking advantage of developments in the Physics package during 2018, below we show the computation up to  and provide a compact approach to compute them up to arbitrary finite order.

Computing up to

Set the signature of spacetime such that its space part is equal to +++ and use lowercaselatin letters to represent space indices. Set also ,  and  to represent quantum operators

 >
 >
 (1)

To illustrate the computation up to , a convenient example, where the commutator algebra is closed, consists of taking  and  as Pauli Matrices which, multiplied by the imaginary unit, form a basis for the group, which in turn exponentiate to the relevant Special Unitary Group . The algebra for the Pauli matrices involves a commutator and an anticommutator

 >
 (2)

Assign now  and  to two Pauli matrices, for instance

 >
 (3)
 >
 (4)

Next, to extract the coefficient of  from

to solve it for  we note that each term has a factor  multiplying a sum, so we only need to take into account the first  terms (sums) and in each sum replace  by the corresponding . For example, given to compute  we only need to compute these first three terms:

then solving for  one gets .

Also, since to compute  we only need the coefficient of , it is not necessary to compute all the terms of each multiple-sum. One way of restricting the multiple-sums to only one power of  consists of using multi-index summation, available in the Physics package (see Physics:-Library:-Add ). For that purpose, redefine sum to extend its functionality with multi-index summation

 >
 (5)

Now we can represent the same computation of  without multiple sums and without computing unnecessary terms as

Finally, we need a computational representation for the repeated commutator bracket

One way of representing this commutator bracket operation is defining a procedure, say F, with a cache to avoid recomputing lower order nested commutators, as follows

 >
 (6)
 >

For example,

 >
 (7)
 >
 (8)
 >
 (9)

We can set now the value of

 >
 (10)

and enter the formula that involves only multi-index summation

 >
 (11)

from where we compute  by solving for it the coefficient of , and since due to the mulit-index summation this expression already contains  as a factor,

 >
 (12)

In order to generalize the formula for H for higher powers of , the right-hand side of the multi-index summation limit can be expressed in terms of an abstract N, and H transformed into a mapping:

 >
 (13)

Now we have

 >
 (14)
 >
 (15)

The following is already equal to (11)

 >
 (16)

In this way, we can reproduce the results published in the literature for the coefficients of Zassenhaus formula up to  by adding two more multi-index sums to (13). Unassign  first

 >
 >

We compute now up to  in one go

 >
 (17)

The nested-commutator expression solved in the last step for  is

 >
 (18)

With everything understood, we want now to extend these results generalizing them into an approach to compute an arbitrarily large coefficient , then use that generalization to compute all the Zassenhaus coefficients up to . To type the formula for H for higher powers of  is however prone to typographical mistakes. The following is a program, using the Maple programming language , that produces these formulas for an arbitrary integer power of :

Formula := proc(A, B, C, Q)

This Formula program uses a sequence of summation indices with as much indices as the order of the coefficient  we want to compute, in this case we need 10 of them

 >
 (19)

To avoid interference of the results computed in the loop (17), unassign  again

 >

Now the formulas typed by hand, used lines above to compute each of ,  and , are respectively constructed by the computer

 >
 (20)
 >
 (21)
 >
 (22)

Construct then the formula for  and make it be a mapping with respect to N, as done for  after (16)

 >
 (23)

Compute now the coefficients of the Zassenhaus formula up to  all in one go

 >
 (24)

Notes: with the material above you can compute higher order values of . For that you need:

 1 Unassign  as done above in two opportunities, to avoid interference of the results just computed.
 2 Indicate more summation indices in the sequence  in (19), as many as the maximum value of n in .
 3 Have in mind that the growth in size and complexity is significant, with each  taking significantly more time than the computation of all the previous ones.
 4 Re-execute the input line (23) and the loop (24).
 >

Download The_Zassenhause_formula_and_the_Pauli_Matrices.mw

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

 5 6 7 8 9 10 11 Last Page 7 of 48
﻿