ecterrab

ACA 2017 - Computer Algebra in High-Scho...

Posted: ecterrab 14540 Product: Maple

August 02 2017

3 0

I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This first presentation, about "Active Learning in High-School Mathematics using Interactive Interfaces", describes a project I started working 23 years ago, which I believe will be part of the future in one or another form. This is work actually not related to my work at Maplesoft.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

Active learning in High-School mathematics using Interactive Interfaces

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Abstract:

The key idea in this project is to learn through exploration using a web of user-friendly Highly Interactive Graphical Interfaces (HIGI). The HIGIs, structured as trees of interlinked windows, present concepts using a minimal amount of text while maximizing the possibility of visual and analytic exploration. These interfaces run computer algebra software in the background. Assessment tools are integrated into the learning experience within the general conceptual map, the Navigator. This Navigator offers students self-assessment tools and full access to the logical sequencing of course concepts, helping them to identify any gaps in their knowledge and to launch the corresponding learning interfaces. An interactive online set of HIGIS of this kind can be used at school, at home, in distance education, and both individually and in a group.

	Computer algebra interfaces for High-School students of "Colegio de Aplicação" (UERJ/1994)

Motivation

When we are the average high-school student facing mathematics, we tend to feel

•	Bored, fragmentarily taking notes, listening to a teacher for 50 or more minutes

•	Anguished because we do not understand some math topics (too many gaps accumulated)

•	Powerless because we don't know what to do to understand (don't have any instant-tutor to ask questions and without being judged for having accumulated gaps)

•	Stressed by the upcoming exams where the lack of understanding may become evident

Computer algebra environments can help in addressing these issues.

•	Be as active as it can get while learning at our own pace.

•	Explore at high speed and without feeling judged. There is space for curiosity with no computational cost.

•	Feel empowered by success. That leads to understanding.

•	Possibility for making of learning a social experience.

Interactive interfaces

Interactive interfaces do not replace the teacher - human learning is an emotional process. A good teacher leading good active learning is a positive experience a student will never forget

Not every computer interface is a valuable resource, at all. It is the set of pedagogical ideas implemented that makes an interface valuable (the same happens with textbooks)

A course on high school mathematics using interactive interfaces - the Edukanet project

–	Brazilian and Canadian students/programmers were invited to participate - 7 people worked in the project.

–	Some funding provided by the Brazilian Research agency CNPq.

Tasks:

-Develop a framework to develop the interfaces covering the last 3 years of high school mathematics (following the main math textbook used in public schools in Brazil)

- Design documents for the interfaces according to given pedagogical guidelines.

- Create prototypes of Interactive interfaces, running Maple on background, according to design document and specified layout (allow for everybody's input/changes).

	The pedagogical guidelines for interactive interfaces

The Math-contents design documents for each chapter

	Example: complex numbers

Each math topic: a interactive interrelated interfaces (windows)

For each topic of high-school mathematics (chapter of a textbook), develop a tree of interactive interfaces (applets) related to the topic (main) and subtopics

Example: Functions

•	Main window

•	Analysis window

•

•	Parity window

•	Visualization of function's parity

•	Step-by-Step solution window

The Navigator: a window with a tile per math topic

•	Click the topic-tile to launch a smaller window, topic-specific, map of interrelated sub-topic tiles, that indicates the logical sequence for the sub-topics, and from where one could launch the corresponding sub-topic interactive interface.

•	This topic-specific smaller window allows for identifying the pre-requisites and gaps in understanding, launching the corresponding interfaces to fill the gaps, and tracking the level of familiarity with a topic.

	The framework to create the interfaces: a version of NetBeans on steroids ...

Complementary classroom activity on a computer algebra worksheet

This course is organized as a guided experience, 2 hours per day during five days, on learning the basics of the Maple language, and on using it to formulate algebraic computations we do with paper and pencil in high school and 1st year of undergraduate science courses.

Explore. Having success doesn't matter, using your curiosity as a compass does - things can be done in so many different ways. Have full permission to fail. Share your insights. All questions are valid even if to the side. Computer algebra can transform the learning of mathematics into interesting understanding, success and fun.

	1. Arithmetic operations and elementary functions

	2. Algebraic Expressions, Equations and Functions

	3. Limits, Derivatives, Sums, Products, Integrals, Differential Equations

	4. Algebraic manipulation: simplify, factorize, expand

	5. Matrices (Linear Algebra)

	Advanced students: guiding them to program mathematical concepts on a computer algebra worksheet

Status of the project

Prototypes of interfaces built cover:

•	Natural numbers

•

Functions

•	Integer numbers

•	Rational numbers

•	Absolute value

•	Logarithms

•	Numerical sequences

•	Trigonometry

•

Matrices

•	Determinants

•	Linear systems

•

Limits

•	Derivatives

•	Derivative of the inverse function

•	The point in Cartesian coordinates

•

The line

•	The circle

•	The ellipse

•	The parabole

•	The hyperbole

•	The conics

	More recent computer algebra frameworks: Maple Mobius for online courses and automated evaluation

Download Computer_Algebra_in_Education.mw

Download Computer_Algebra_in_Education.pdf

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

Physics in Maple 2017: Special, General ...

Posted: ecterrab 14540 Product: Maple 2017

June 15 2017

3 9

Physics

Maple provides a state-of-the-art environment for algebraic and tensorial computations in Physics, with emphasis on ensuring that the computational experience is as natural as possible.

The theme of the Physics project for Maple 2017 has been the consolidation of the functionality introduced in previous releases, together with significant enhancements and new functionality in General Relativity, in connection with classification of solutions to Einstein's equations and tensor representations to work in an embedded 3D curved space - a new ThreePlusOne package. This package is relevant in numerical relativity and a Hamiltonian formulation of gravity. The developments also include first steps in connection with computational representations for all the objects entering the Standard Model in particle physics.

Classification of solutions to Einstein's equations and the Tetrads package

In Maple 2016, the digitizing of the database of solutions to Einstein's equations was finished, added to the standard Maple library, with all the metrics from "Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E., Exact Solutions to Einstein's Field Equations". These metrics can be loaded to work with them, or change them, or searched using g_ (the Physics command representing the spacetime metric that also sets the metric to your choice in one go) or using the command DifferentialGeometry:-Library:-MetricSearch .

In Maple 2017, the Physics:-Tetrads package has been vastly improved and extended, now including new commands like PetrovType and SegreType to classify these metrics, and the TransformTetrad now has an option canonicalform to automatically derive a transformation and put the tetrad in canonical form (reorientation of the axis of the local system of references), a relevant step in resolving the equivalence between two metrics.

Examples

	Petrov and Segre types, tetrads in canonical form

Equivalence for Schwarzschild metric (spherical and Kruskal coordinates)

	Formulation of the problem (remove mixed coordinates)

	Solving the Equivalence

The ThreePlusOne (3 + 1) new Maple 2017 Physics package

ThreePlusOne , is a package to cast Einstein's equations in a 3+1 form, that is, representing spacetime as a stack of nonintersecting 3-hypersurfaces Σ. This description is key in the Hamiltonian formulation of gravity as well as in the study of gravitational waves, black holes, neutron stars, and in general to study the evolution of physical system in general relativity by running numerical simulations as traditional initial value (Cauchy) problems. ThreePlusOne includes computational representations for the spatial metric that is induced by on the 3-dimensional hypersurfaces, and the related covariant derivative, Christoffel symbols and Ricci and Riemann tensors, the Lapse, Shift, Unit normal and Time vectors and Extrinsic curvature related to the ADM equations.

The following is a list of the available commands:

ADMEquations	Christoffel3	D3_	ExtrinsicCurvature
gamma3_	Lapse	Ricci3	Riemann3
Shift	TimeVector	UnitNormalVector

The other four related new Physics commands:

•	Decompose , to decompose 4D tensorial expressions (free and/or contracted indices) into the space and time parts.

•	gamma_ , representing the three-dimensional metric tensor, with which the element of spatial distance is defined as .

•	Redefine , to redefine the coordinates and the spacetime metric according to changes in the signature from any of the four possible signatures(− + + +), (+ − − −), (+ + + −) and ((− + + +) to any of the other ones.

•	EnergyMomentum , is a computational representation for the energy-momentum tensor entering Einstein's equations as well as their 3+1 form, the ADMEquations .

Examples

>

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (x, y, z, t)}$

(2.1.1)

>

0, "%1 is not a command in the %2 package", ThreePlusOne, Physics

(2.1.2)

Note the different color for , now a 4D tensor representing the metric of a generic 3-dimensional hypersurface induced by the 4D spacetime metric . All the ThreePlusOne tensors are displayed in black to distinguish them of the corresponding 4D or 3D tensors. The particular hypersurface operates is parameterized by the Lapse and the Shift .

The induced metric is defined in terms of the UnitNormalVector and the 4D metric as

>

(2.1.3)

where is defined in terms of the Lapse and the derivative of a scalar function t that can be interpreted as a global time function

>

(2.1.4)

The TimeVector is defined in terms of the Lapse and the Shift and this vector as

>

(2.1.5)

The ExtrinsicCurvature is defined in terms of the LieDerivative of

>

Physics:-ThreePlusOne:-ExtrinsicCurvature[mu, nu] = -(1/2)*Physics:-LieDerivative[Physics:-ThreePlusOne:-UnitNormalVector](Physics:-ThreePlusOne:-gamma3_[mu, nu])

(2.1.6)

The metric is also a projection tensor in that it projects 4D tensors into the 3D hypersurface Σ. The definition for any 4D tensor that is also a 3D tensor in Σ, can thus be written directly by contracting their indices with . In the case of Christoffel3 , Ricci3 and Riemann3, these tensors can be defined by replacing the 4D metric by and the 4D Christoffel symbols by the ThreePlusOne in the definitions of the corresponding 4D tensors. So, for instance

>

Physics:-ThreePlusOne:-Christoffel3[mu, nu, alpha] = (1/2)*Physics:-ThreePlusOne:-gamma3_[mu, `~beta`]*(Physics:-d_[alpha](Physics:-ThreePlusOne:-gamma3_[beta, nu], [X])+Physics:-d_[nu](Physics:-ThreePlusOne:-gamma3_[beta, alpha], [X])-Physics:-d_[beta](Physics:-ThreePlusOne:-gamma3_[nu, alpha], [X]))

(2.1.7)

>

(2.1.8)

>

(2.1.9)

When working with the ADM formalism, the line element of an arbitrary spacetime metric can be expressed in terms of the differentials of the coordinates , the Lapse , the Shift and the spatial components of the 3D metric gamma3_ . From this line element one can derive the relation between the Lapse , the spatial part of the Shift , the spatial part of the gamma3_ metric and the components of the 4D spacetime metric.

For this purpose, define a tensor representing the differentials of the coordinates and an alias

>

${Physics:-ThreePlusOne:-D3_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-ThreePlusOne:-Ricci3[mu, nu], Physics:-ThreePlusOne:-Shift[mu], Physics:-d_[mu], dx[mu], Physics:-g_[mu, nu], Physics:-ThreePlusOne:-gamma3_[mu, nu], Physics:-gamma_[i, j], Physics:-ThreePlusOne:-Christoffel3[mu, nu, alpha], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-ThreePlusOne:-Riemann3[mu, nu, alpha, beta], Physics:-ThreePlusOne:-TimeVector[mu], Physics:-ThreePlusOne:-ExtrinsicCurvature[mu, nu], Physics:-ThreePlusOne:-UnitNormalVector[mu], Physics:-SpaceTimeVector[mu](X)}$

(2.1.10)

>

The expression for the line element in terms of the Lapse and Shift is (see [2], eq.(2.123))

>

(2.1.11)

Compare this expression with the 3+1 decomposition of the line element in an arbitrary system. To avoid the automatic evaluation of the metric components, work with the inert form of the metric %g_

>

(2.1.12)

>

(2.1.13)

The second and third terms on the right-hand side are equal

>

(2.1.14)

>

(2.1.15)

Taking the difference between this expression and the one in terms of the Lapse and Shift we get

>

(2.1.16)

Taking coefficients, we get equations for the Shift , the Lapse and the spatial components of the metric gamma3_

>

(2.1.17)

>

(2.1.18)

>

(2.1.19)

Using these equations, these quantities can all be expressed in terms of the time and space components of the 4D metric and

>

(2.1.20)

>

(2.1.21)

>

(2.1.22)

References

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

[2] Alcubierre, M., Introduction to 3+1 Numerical Relativity, International Series of Monographs on Physics 140, Oxford University Press, 2008.

[3] Baumgarte, T.W., Shapiro, S.L., Numerical Relativity, Solving Einstein's Equations on a Computer, Cambridge University Press, 2010.

[4] Gourgoulhon, E., 3+1 Formalism and Bases of Numerical Relativity, Lecture notes, 2007, https://arxiv.org/pdf/gr-qc/0703035v1.pdf.

[5] Arnowitt, R., Dese, S., Misner, C.W., The Dynamics of General Relativity, Chapter 7 in Gravitation: an introduction to current research (Wiley, 1962), https://arxiv.org/pdf/gr-qc/0405109v1.pdf

Examples: Decompose, gamma_

>

(2.2.1)

Define now an arbitrary tensor

>

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

(2.2.2)

So is a 4D tensor with only one free index, where the position of the time-like component is the position of the different sign in the signature, that you can query about via

>

(2.2.3)

To perform a decomposition into space and time, set - for instance - the lowercase latin letters from i to s to represent spaceindices and

>

(2.2.4)

Accordingly, the 3+1 decomposition of is

>

(2.2.5)

The 3+1 decomposition of the inert representation %g_[mu,nu] of the 4D spacetime metric; use the inert representation when you do not want the actual components of the metric appearing in the output

>

Matrix(%id = 18446744078724507998)

(2.2.6)

Note the position of the component %g_[0, 0], related to the trailing position of the time-like component in the signature .

Compare the decomposition of the 4D inert with the decomposition of the 4D active spacetime metric

>

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))

(2.2.7)

>

Matrix(%id = 18446744078724494270)

(2.2.8)

Note that in general the 3D space part of is not equal to the 3D metric whose definition includes another term (see [1] Landau & Lifshitz, eq.(84.7)).

>

Physics:-gamma_[i, j] = -Physics:-g_[i, j]+Physics:-g_[0, i]*Physics:-g_[0, j]/Physics:-g_[0, 0]

(2.2.9)

The 3D space part of is actually equal to the 3D metric

>

(2.2.10)

To derive the formula for the covariant components of the 3D metric, Decompose into 3+1 the identity

>

(2.2.11)

To the side, for illustration purposes, these are the 3 + 1 decompositions, first excluding the repeated indices, then excluding the free indices

>

Matrix(%id = 18446744078132963318)

(2.2.12)

>

(2.2.13)

Compare with a full decomposition

>

Matrix(%id = 18446744078724489454)

(2.2.14)

is a symmetric matrix of equations involving non-contracted occurrences of , and . Isolate, in , , that you input as %g_[~j, ~0], and substitute into

>

%g_[`~j`, `~0`] = -%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]

(2.2.15)

>

-%g_[0, k]*%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]+%g_[i, k]*%g_[`~j`, `~i`] = Physics:-KroneckerDelta[k, `~j`]

(2.2.16)

Collect , that you input as %g_[~j, ~i]

>

(-%g_[0, k]*%g_[i, 0]/%g_[0, 0]+%g_[i, k])*%g_[`~j`, `~i`] = Physics:-KroneckerDelta[k, `~j`]

(2.2.17)

Since the right-hand side is the identity matrix and, from , , the expression between parenthesis, multiplied by -1, is the reciprocal of the contravariant 3D metric , that is the covariant 3D metric , in accordance to its definition for the signature

>

Physics:-gamma_[i, j] = -Physics:-g_[i, j]+Physics:-g_[0, i]*Physics:-g_[0, j]/Physics:-g_[0, 0]

(2.2.18)

>

References

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

	Example: Redefine

Tensors in Special and General Relativity

A number of relevant changes happened in the tensor routines of the Physics package, towards making the routines pack more functionality both for special and general relativity, as well as working more efficiently and naturally, based on Maple's Physics users' feedback collected during 2016.

New functionality

•	Implement conversions to most of the tensors of general relativity (relevant in connection with functional differentiation)

•	New setting in the Physics Setup allows for specifying the cosmologicalconstant and a default tensorsimplifier

Download PhysicsMaple2017.mw

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

Digitizing mathematics: ODEs, Special Fu...

Posted: ecterrab 14540 Product: Maple

October 03 2016

4 3

The material below was presented in the "Semantic Representation of Mathematical Knowledge Workshop", February 3-5, 2016 at the Fields Institute, University of Toronto. It shows the approach I used for “digitizing mathematical knowledge" regarding Differential Equations, Special Functions and Solutions to Einstein's equations. While for these areas using databases of information helps (for example textbooks frequently contain these sort of databases), these are areas that, at the same time, are very suitable for using algorithmic mathematical approaches, that result in much richer mathematics than what can be hard-coded into a database. The material also focuses on an interesting cherry-picked collection of Maple functionality, that I think is beautiful, not well know, and seldom focused inter-related as here.

Digitizing of special functions,

differential equations,

and solutions to Einstein’s equations

within a computer algebra system

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Editor, Computer Physics Communications

Digitizing (old paradigm)

•	Big amounts of knowledge available to everybody in local machines or through the internet

•	Take advantage of basic computer functionality, like searching and editing

Digitizing (new paradigm)

•	By digitizing mathematical knowledge inside appropriate computational contexts that understand about the topics, one can use the digitized knowledge to automatically generate more and higher level knowledge

Challenges

1) how to identify, test and organize the key blocks of information,

2) how to access it: the interface,

3) how to mathematically process it to automatically obtain more information on demand

Three examples

Mathematical Functions

"Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

The FunctionAdvisor (basic)

"Supporting information on definitions, identities, possible simplifications, integral forms, different types of series expansions, and mathematical properties in general"

	Examples

	General description

	References

Differential equation representation for generic nonlinear algebraic expressions - their use

"Compute differential polynomial forms for arbitrary systems of non-polynomial equations ..."

	The Differential Equations representing arbitrary algebraic expresssions

	Deriving knowledge: ODE solving methods

	Extending the mathematical language to include the inverse functions

	Solving non-polynomial algebraic equations by solving polynomial differential equations

	References

Branch Cuts of algebraic expressions

"Algebraically compute, and visualize, the branch cuts of arbitrary mathematical expressions"

	Examples

	References

Algebraic expresssions in terms of specified functions

"A conversion network for arbitrary mathematical expressions, to rewrite them in terms of different functions in flexible ways"

	Examples

	General description

	References

Symbolic differentiation of algebraic expressions

"Perform symbolic differentiation by combining different algebraic techniques, including functions of symbolic sequences and Faà di Bruno's formula"

	Examples

	References

Ordinary Differential Equations

"Beyond the concept of a database, classify an arbitrary ODE and suggest solution methods for it"

	General description

	Examples

	References

Exact Solutions to Einstein's equations

"The authors of "Exact solutions toEinstein's equations" reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, organized the whole material into chapters according to the physical properties of these solutions. These solutions are key in the area of general relativity, are now all digitized and become alive in a worksheet"

The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords (old paradigm) changes the game.

More important, within a computer algebra system this knowledge becomes alive (new paradigm).

•	The solutions are turned active by a simple call to one commend, called the g_ spacetime metric.

•	Everything else gets automatically derived and set on the fly ( Christoffel symbols , Ricci and Riemann tensors orthonormal and null tetrads , etc.)

•	Almost all of the mathematical operations one can perform on these solutions are implemented as commands in the Physics and DifferentialGeometry packages.

•	All the mathematics within the Maple library are instantly ready to work with these solutions and derived mathematical objects.

Finally, in the Maple PDEtools package , we have all the mathematical tools to tackle the equivalence problem around these solutions.

	Examples

	References

Download: Digitizing_Mathematical_Information.mw, Digitizing_Mathematical_Information.pdf

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

Quantization of the Lorentz Force...

Posted: ecterrab 14540 Product: Maple

September 26 2016

4 0

This presentation is on an undergrad intermediate Quantum Mechanics topic. Tackling the problem within a computer algebra worksheet in the way shown below is actually the novelty, using the Physics package to formulate the problem with quantum operators and related algebra rules in tensor notation.

Quantization of the Lorentz Force

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab²

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

We consider the case of a quantum, non-relativistic, particle with mass m and charge q evolving under the action of an arbitrary time-independent magnetic field where is the vector potential. The Hamiltonian for this system is

H = (`#mover(mi("p",mathcolor = "olive"),mo("→"))`-q*`#mover(mi("A",mathcolor = "olive"),mo("→"))`(X))^2/(2*m)

where is the momentum of the particle, and the force acting in this particle, also called the Lorentz force, is given by

where is the quantized velocity of the particle, and all of , , , , and are Hermitian quantum operators representing observable quantities.

In the classic (non-quantum) case, for such a particle in the absence of electrical field is given by

= ,

Problem: Departing from the Hamiltonian, show that in the quantum case the Lorentz force is given by [1]

[1] Photons et atomes, Introduction à l'électrodynamique quantique, p. 179, Claude Cohen-Tannoudji, Jacques Dupont-Roc et Gilbert Grynberg - EDP Sciences janvier 1987.

Solution

We choose to tackle the problem in Heisenberg's picture of quantum mechanices, where the state of a system is static and only the quantum operators evolve in time according to

Also, the algebraic manipulations are simpler using tensor abstract notation instead of the standard 3D vector notation. We then start setting the framework for the problem, a system of coordinates X, indicating the dimension of the tensor space to be 3 and the metric Euclidean, and that we will use lowercaselatin letters to represent tensor indices. In addition, not necessary but for convenience, we set the lowercase latin i to represent the imaginary unit and we request automaticsimplification so that the output of everything comes automatically simplified in size.

>

(1)

Next we indicate the letters we will use to represent the quantum operators with which we will work, and also the standard commutation rules between position and momentum, always the starting point when dealing with quantum mechanics problems

>

Setup(quantumoperators = {F}, hermitianoperators = {A, B, H, p, r, v, x}, realobjects = {`&hbar;`, m, q}, algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(x[k], p[l]) = I*`&hbar;`*KroneckerDelta[k, l], %Commutator(x[k], x[l]) = 0})

[algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(x[k], p[l]) = I*`&hbar;`*Physics:-KroneckerDelta[k, l], %Commutator(x[k], x[l]) = 0}, hermitianoperators = {A, B, H, p, r, v, x}, quantumoperators = {A, B, F, H, p, r, v, x}, realobjects = {`&hbar;`, m, q, x1, x2, x3, %dAlembertian, Physics:-dAlembertian}]

(2)

Note that we start not indicating as Hermitian, in order to arrive at that result. The quantum operators , , and are explicit functions of X, so to avoid redundant display of this functionality on the screen we use

>

(3)

Define now as tensors the quantum operators that we will use with tensorial notation (recalling: for these, Einstein's sum rule for repeated indices will be automatically applied when simplifying)

>

${A, B, F, p, v, x, Physics:-Dgamma[a], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(4)

The Hamiltonian,

H = (`#mover(mi("p",mathcolor = "olive"),mo("→"))`-q*`#mover(mi("A",mathcolor = "olive"),mo("→"))`(X))^2/(2*m)

in tensorial notation, is given by

>

H = (1/2)*Physics:-`^`(p[n]-q*A[n](X), 2)/m

(5)

Generally speaking to arrive at what we now need to do is

1) Express this Hamiltonian (5) in terms of the velocity

And, recalling that, in Heisenberg's picture, quantum operators evolve in time according to

2) Take the commutator of with the velocity itself to obtain its time derivative and, from , that commutator is already the force up to some constant factors.

To get in contact with the basic commutation rules between position and momentum behind quantum phenomena, the quantized velocity itself can be computed as the time derivative of the position operator, i.e as the commutator of with

>

I*Physics:-Commutator(H, x[k])/`&hbar;` = (1/2)*(I*q^2*Physics:-AntiCommutator(A[n](X), Physics:-Commutator(A[n](X), x[k]))-I*q*Physics:-AntiCommutator(p[n], Physics:-Commutator(A[n](X), x[k]))-2*(q*A[n](X)-p[n])*Physics:-KroneckerDelta[k, n]*`&hbar;`)/(`&hbar;`*m)

(6)

This expression for the velocity, that involves commutators between the potential , the position and the momentum , can be simplified taking into account the basic quantum algebra rules between position and momentum. We assume that (X) can be decomposed into a formal power series (possibly infinite) of the , hence all the commute between themselves as well as with all the

>

(7)

(Note: in some cases, this is not true, but those cases are beyond the scope of this worksheet.)

Add these rules to the algebra rules already set so that they are all taken into account when simplifying things

>

(8)

>

I*Physics:-Commutator(H, x[k])/`&hbar;` = (-A[k](X)*q+p[k])/m

(9)

The right-hand side of (9) is then the k^th component of the velocity tensor quantum operator, the relationship is the same as in the classical case

>

(10)

and with this the Hamiltonian (5) can now be rewritten in term of the velocity completing step 1)

>	$simplify(H = (1/2)Physics[`^`](p[n]-qA[n](X), 2)/m, {SubstituteTensorIndices(k = n, (rhs = lhs)(v[k] = (-A[k](X)*q+p[k])/m))})$

H = (1/2)*m*Physics:-`^`(v[n], 2)

(11)

For step 2), to compute

`#mover(mi("F",mathcolor = "olive"),mo("→"))` = m*(diff(v(t), t)) and m*(diff(v(t), t)) = I*m*Physics:-Commutator(H, v(t)[k])/`&hbar;`

we need the commutator between the different components of the quantized velocity which, contrary to what happens in the classical case, do not commute. For this purpose, take the commutator between (10) with itself after replacing the free index

>

Physics:-Commutator(v[k], v[n]) = -q*(Physics:-Commutator(A[k](X), p[n])+Physics:-Commutator(p[k], A[n](X)))/m^2

(12)

To simplify (12), we use the fact that if f is a commutative mapping that can be decomposed into a formal power series in all the complex plan (which is assumed to be the case for all (X)), then

where is the momentum operator along the axis. This relation reads in tensor notation:

>

(13)

Add this rule to the rules previously set in order to automatically take it into account in (12)

>

$[algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(p[k], A[n](X)) = -I*`&hbar;`*Physics:-d_[k](A[n](X), [X]), %Commutator(x[k], p[l]) = I*`&hbar;`*Physics:-KroneckerDelta[k, l], %Commutator(x[k], x[l]) = 0, %Commutator(A[k](X), x[l]) = 0, %Commutator(A[k](X), A[l](X)) = 0}]$

(14)

>

Physics:-Commutator(v[k], v[n]) = -I*q*`&hbar;`*(Physics:-d_[n](A[k](X), [X])-Physics:-d_[k](A[n](X), [X]))/m^2

(15)

Also add this other rule so that it is taken into account automatically

>

$[algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(p[k], A[n](X)) = -I*`&hbar;`*Physics:-d_[k](A[n](X), [X]), %Commutator(v[k], v[n]) = -I*q*`&hbar;`*(Physics:-d_[n](A[k](X), [X])-Physics:-d_[k](A[n](X), [X]))/m^2, %Commutator(x[k], p[l]) = I*`&hbar;`*Physics:-KroneckerDelta[k, l], %Commutator(x[k], x[l]) = 0, %Commutator(A[k](X), x[l]) = 0, %Commutator(A[k](X), A[l](X)) = 0}]$

(16)

Recalling now the expression of the Hamiltonian (11) as a function of the velocity, one can compute the components of the force operator "()Component(v*B,k)=m (v[k])=(i m [H,v[k]][-])/`&hbar;`"

>

F[k](X) = I*m*%Commutator((1/2)*m*Physics:-`^`(v[n], 2), v[k])/`&hbar;`

(17)

Simplify this expression for the quantized force taking the quantum algebra rules (16) into account

>

F[k](X) = (1/2)*q*(-Physics:-`*`(Physics:-d_[n](A[k](X), [X]), v[n])+Physics:-`*`(Physics:-d_[k](A[n](X), [X]), v[n])-Physics:-`*`(v[n], Physics:-d_[n](A[k](X), [X]))+Physics:-`*`(v[n], Physics:-d_[k](A[n](X), [X])))

(18)

It is not difficult to verify that this is the antisymmetrized vector product . Departing from expressed using tensor notation,

>

(19)

and taking into acount that

multiply both sides of (19) by , getting

>

(20)

>

(21)

Finally, replacing the repeated index m by n

>

(22)

Likewise, for

multiplying (19), this time from the right instead of from the left, we get

>

(23)

>

(24)

Simplifying now the expression (18) for the quantized force taking into account (22) and (24) we get

>

$simplify(F[k](X) = (1/2)*q*(-Physics[`*`](Physics[d_][n](A[k](X), [X]), v[n])+Physics[`*`](Physics[d_][k](A[n](X), [X]), v[n])-Physics[`*`](v[n], Physics[d_][n](A[k](X), [X]))+Physics[`*`](v[n], Physics[d_][k](A[n](X), [X]))), {(rhs = lhs)(Physics[LeviCivita][b, c, k]*Physics[`*`](v[b], B[c](X)) = Physics[`*`](v[n], Physics[d_][k](A[n](X), [X]))-Physics[`*`](v[n], Physics[d_][n](A[k](X), [X]))), (rhs = lhs)(Physics[LeviCivita][b, c, k]*Physics[`*`](B[c](X), v[b]) = Physics[`*`](Physics[d_][k](A[n](X), [X]), v[n])-Physics[`*`](Physics[d_][n](A[k](X), [X]), v[n]))})$

F[k](X) = (1/2)*q*Physics:-LeviCivita[b, c, k]*(Physics:-`*`(v[b], B[c](X))+Physics:-`*`(B[c](X), v[b]))

(25)

i.e.

in tensor notation. Finally, we note that this operator is Hermitian as expected

>

(26)

Download: Quantization_of_the_Lorentz_force.mw, Quantization_of_the_Lorentz_force.pdf

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

Quantum Mechanics: Schrödinger vs Heisen...

Posted: ecterrab 14540 Product: Maple

September 01 2016

4 0

The presentation below is on undergrad Quantum Mechanics. Tackling this topic within a computer algebra worksheet in the way it's done below, however, is an exciting novelty and illustrates well the level of abstraction that is now possible using the Physics package.

Quantum Mechanics: Schrödinger vs Heisenberg picture

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab²

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

Within the Schrödinger picture of Quantum Mechanics, the time evolution of the state of a system, represented by a Ket , is determined by Schrödinger's equation:

where H, the Hamiltonian, as well as the quantum operators representing observable quantities, are all time-independent.

Within the Heisenberg picture, a Ket representing the state of the system does not evolve with time, but the operators representing observable quantities, and through them the Hamiltonian H, do.

Problem: Departing from Schrödinger's equation,

a) Show that the expected value of a physical observable in Schrödinger's and Heisenberg's representations is the same, i.e. that

b) Show that the evolution equation of an observable in Heisenberg's picture, equivalent to Schrödinger's equation, is given by:

diff(O__H(t), t) = (-I*Physics:-Commutator(O__H(t), H))*(1/`&hbar;`)

where in the right-hand-side we see the commutator of with the Hamiltonian of the system.

Solution: Let and respectively be operators representing one and the same observable quantity in Schrödinger's and Heisenberg's pictures, and H be the operator representing the Hamiltonian of a physical system. All of these operators are Hermitian. So we start by setting up the framework for this problem accordingly, including that the time t and Planck's constant are real. To automatically combine powers of the same base (happening frequently in what follows) we also set combinepowersofsamebase = true. The following input/output was obtained using the latest Physics update (Aug/31/2016) distributed on the Maplesoft R&D Physics webpage.

>

>	$Physics:-Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, t}, combinepowersofsamebase = true, mathematicalnotation = true)$

$[combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, t}]$

(1)

Let's consider Schrödinger's equation

>

(2)

Now, H is time-independent, so (2) can be formally solved: is obtained from the solution at time , as follows:

>

(3)

>

(4)

To check that (4) is a solution of (2), substitute it in (2):

>

(5)

Next, to relate the Schrödinger and Heisenberg representations of an Hermitian operator O representing an observable physical quantity, recall that the value expected for this quantity at time t during a measurement is given by the mean value of the corresponding operator (i.e., bracketing it with the state of the system ).

So let be an observable in the Schrödinger picture: its mean value is obtained by bracketing the operator with equation (4):

>

(6)

The composed operator within the bracket on the right-hand-side is the operator O in Heisenberg's picture,

>

(7)

Analogously, inverting this equation,

>

(8)

As an aside to the problem, we note from these two equations, and since the operator is unitary (because H is Hermitian), that the switch between Schrödinger's and Heisenberg's pictures is accomplished through a unitary transformation.

Inserting now this value of from (8) in the right-hand-side of (6), we get the answer to item a)

>

(9)

where, on the left-hand-side, the Ket representing the state of the system is evolving with time (Schrödinger's picture), while on the the right-hand-side the Ket is constant and it is , the operator representing an observable physical quantity, that evolves with time (Heisenberg picture). As expected, both pictures result in the same expected value for the physical quantity represented by .

To complete item b), the derivation of the evolution equation for , we take the time derivative of the equation (7):

>

diff(O__H(t), t) = I*Physics:-`*`(H, exp(I*t*H/`&hbar;`), O__S, exp(-I*t*H/`&hbar;`))/`&hbar;`-I*Physics:-`*`(exp(I*t*H/`&hbar;`), O__S, H, exp(-I*t*H/`&hbar;`))/`&hbar;`

(10)

To rewrite this equation in terms of the commutator , it suffices to re-order the product H placing the exponential first:

>

diff(O__H(t), t) = I*Physics:-`*`(exp(I*t*H/`&hbar;`), H, O__S, exp(-I*t*H/`&hbar;`))/`&hbar;`-I*Physics:-`*`(exp(I*t*H/`&hbar;`), Physics:-`*`(H, O__S)+Physics:-Commutator(O__S, H), exp(-I*t*H/`&hbar;`))/`&hbar;`