ecterrab

Integral Transforms (revamped) and PDE...

Posted: ecterrab 14540 Product: Maple

October 01 2019

8 14

Integral Transforms (revamped) and PDEs

Integral transforms, implemented in Maple as the inttrans package, are special integrals that appear frequently in mathematical-physics and that have remarkable properties. One of the main uses of integral transforms is for the computation of exact solutions to ordinary and partial differential equations with initial/boundary conditions. In Maple, that functionality is implemented in dsolve/inttrans and in pdsolve/boundary conditions .

During the last months, we have been working heavily on several aspects of these integral transform functions and this post is about that. This is work in progress, in collaboration with Katherina von Bulow.

The integral transforms are represented by the commands of the inttrans package:

>

(1)

Three of these commands, addtable, savetable, and setup (this one is new, only present after installing the Physics Updates) are "administrative" commands while the others are computational representations for integrals. For example,

>

[fourier(a, b, z) = Int(a/exp(I*b*z), b = -infinity .. infinity), MathematicalFunctions:-`with no restrictions on `(a, b, z)]

(2)

>

[mellin(a, b, z) = Int(a*b^(z-1), b = 0 .. infinity), MathematicalFunctions:-`with no restrictions on `(a, b, z)]

(3)

For all the integral transform commands, the first argument is the integrand, the second one is the dummy integration variable of a definite integral and the third one is the evaluation point. (also called transform variable). The integral representation is also visible using the convert network

>

laplace(f(t), t, s) = Int(f(t)*exp(-s*t), t = 0 .. infinity)

(4)

Having in mind the applications of these integral transforms to compute integrals and exact solutions to PDE with boundary conditions, five different aspects of these transforms received further development:

•	Compute Derivatives: Yes or No

•	Numerical Evaluation

•	Two Hankel Transform Definitions

•	More integral transform results

•	Mellin and Hankel transform solutions for Partial Differential Equations with boundary conditions

The project includes having all these tranforms available at user level (not ready), say as FourierTransform for inttrans:-fourier, so that we don't need to input anymore. Related to these changes we also intend to have not return anymore, and return itself instead, unevaluated, so that one can set its value according to the problem/preferred convention (typically 0, 1/2 or 1) and have all the Maple library following that choice.

The material presented in the following sections is reproducible already in Maple 2019 by installing the latest Physics Updates (v.435 or higher),

Compute derivatives: Yes or No.

For historical reasons, previous implementations of these integral transform commands did not follow a standard paradigm of computer algebra: "Given a function , the input should return the derivative of ". The implementation instead worked in the opposite direction: if you were to input the result of the derivative, you would receive the derivative representation. For example, to the input you would receive . This is particularly useful for the purpose of using integral transforms to solve differential equations but it is counter-intuitive and misleading; Maple knows the differentiation rule of these functions, but that rule was not evident anywhere. It was not clear how to compute the derivative (unless you knew the result in advance).

To solve this issue, a new command, setup, has been added to the package, so that you can set "whether or not" to compute derivatives, and the default has been changed to computederivatives = true while the old behavior is obtained only if you input . For example, after having installed the Physics Updates,

>

`The "Physics Updates" version in the MapleCloud is 435 and is the same as the version installed in this computer, created 2019, October 1, 12:46 hours, found in the directory /Users/ecterrab/maple/toolbox/2019/Physics Updates/lib/`

(1.1)

the current settings can be queried via

>

(1.2)

and so differentiating returns the derivative computed

>

(1.3)

while changing this setting to work as in previous releases you have this computation reversed: you input the output (1.3) and you get the corresponding input

>

(1.4)

>

(1.5)

Reset the value of computederivatives

>

(1.6)

>

(1.7)

In summary: by default, derivatives of integral transforms are now computed; if you need to work with these derivatives as in previous releases, you can input . This setting can be changed any time you want within one and the same Maple session, and changing it does not have any impact on the performance of intsolve, dsolve and pdsolve to solve differential equations using integral transforms.

>

Numerical Evaluation

In previous releases, integral transforms had no numerical evaluation implemented. This is in the process of changing. So, for example, to numerically evaluate the inverse laplace transform ( invlaplace command), three different algorithms have been implemented: Gaver-Stehfest, Talbot and Euler, following the presentation by Abate and Whitt, "Unified Framework for Numerically Inverting Laplace Transforms", INFORMS Journal on Computing 18(4), pp. 408–421, 2006.

For example, consider the exact solution to this partial differential equation subject to initial and boundary conditions

>

Note that these two conditions are not entirely compatible: the solution returned cannot be valid for and simultaneously. However, a solution discarding that point does exist and is given by

>

u(x, t) = -invlaplace(exp(-(1/2)*s^(1/2)*t)/s, s, x)+1

(2.1)

Verifying the solution, one condition remains to be tested

>

[0, 0, -invlaplace(exp(-(1/2)*s^(1/2)*t)/s, s, 0)]

(2.2)

Since we now have numerical evaluation rules, we can test that what looks different from 0 in the above is actually 0.

>

-invlaplace(exp(-(1/2)*s^(1/2)*t)/s, s, 0)

(2.3)

Add a small number to the initial value of t to skip the point

>

The default method used is the method of Euler sums and the numerical evaluation is performed as usual using the evalf command. For example, consider

>

The Laplace transform of F is given by

>

(1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2)

(2.4)

and the inverse Laplace transform of LT in inert form is

>

%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, t)

(2.5)

At we have

>

%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1)

(2.6)

>

(2.7)

This result is consistent with the one we get if we first compute the exact form of the inverse Laplace transform at t = 1:

>

%invlaplace((1/2)*2^(1/2)*Pi^(1/2)*exp(-(1/2)/s)/s^(3/2), s, 1) = sin(2^(1/2))

(2.8)

>

(2.9)

In addition to the standard use of evalf to numerically evaluate inverse Laplace transforms, one can invoke each of the three different methods implemented using the MathematicalFunctions:-Evalf command

>

(2.10)

>

(2.11)

>

(2.12)

>

(2.13)

Regarding the method we use by default: from a numerical experiment with varied problems we have concluded that our implementation of the Euler (sums) method is faster and more accurate than the other two.

Two Hankel transform definitions

In previous Maple releases, the definition of the Hankel transform was given by

hankel(f(t), t, s, nu) = Int(f(t)*sqrt(s*t)*BesselJ(nu, s*t), t = 0 .. infinity)

where is the function. This definition, sometimes called alternative definition of the Hankel transform, has the inconvenience of the square root in the integrand, complicating the form of the hankel transform for the Laplacian in cylindrical coordinates. On the other hand, the definition more frequently used in the literature is

hankel(f(t), t, s, nu) = Int(f(t)*t*BesselJ(nu, s*t), t = 0 .. infinity)

With it, the Hankel transform of diff(u(r, t), r, r)+(diff(u(r, t), r))/r+diff(u(r, t), t, t) is given by the simple ODE form d^2*`&Hopf;`(k, t)/dt^2-k^2*`&Hopf;`(k, t) . Not just this transform but several other ones acquire a simpler form with the definition that does not have a square root in the integrand.

So the idea is to align Maple with this simpler definition, while keeping the previous definition as an alternative. Hence, by default, when you load the inttrans package, the new definition in use for the Hankel transform is

>

hankel(f(t), t, s, nu) = Int(f(t)*t*BesselJ(nu, s*t), t = 0 .. infinity)

(3.1)

You can change this default so that Maple works with the alternative definition as in previous releases. For that purpose, use the new inttrans:-setup command (which you can also use to query about the definition in use at any moment):

>

(3.2)

This change in definition is automatically taken into account by other parts of the Maple library using the Hankel transform. For example, the differentiation rule with the new definition is

>

(3.3)

This differentiation rule resembles (is connected to) the differentiation rule for BesselJ, and this is another advantage of the new definition.

>

%diff(BesselJ(nu, z), z) = -BesselJ(nu+1, z)+nu*BesselJ(nu, z)/z

(3.4)

Furthermore, several transforms have acquired a simpler form, as for example:

>

%hankel(exp(I*a*r)/r, r, k, 0) = 1/(-a^2+k^2)^(1/2)

(3.5)

Let's compare: make the definition be as in previous releases.

>

(3.6)

>

hankel(f(t), t, s, nu) = Int(f(t)*(s*t)^(1/2)*BesselJ(nu, s*t), t = 0 .. infinity)

(3.7)

The differentiation rule with the previous (alternative) definition was not as simple:

>

(3.8)

And the transform (3.5) was also not so simple:

>

%hankel(exp(I*a*r)/r, r, k, 0) = (I*a*hypergeom([3/4, 3/4], [3/2], a^2/k^2)*GAMMA(3/4)^4+Pi^2*k*hypergeom([1/4, 1/4], [1/2], a^2/k^2))/(k*Pi*GAMMA(3/4)^2)

(3.9)

Reset to the new default value of the definition.

>

(3.10)

>

hankel(f(t), t, s, nu) = Int(f(t)*t*BesselJ(nu, s*t), t = 0 .. infinity)

(3.11)

More integral transform results

The revision of the integral transforms includes also filling gaps: previous transforms that were not being computed are now computed. Still with the Hankel transform, consider the operators

>

`D/t` := proc (u) options operator, arrow; (diff(u, t))/t end proc

Being able to transform these operators into algebraic expressions or differential equations of lower order is key for solving PDE problems with Boundary Conditions.

Consider, for instance, this ODE

>

(4.1)

>

((diff(diff(diff(u(t), t), t), t))*t^3-12*(diff(diff(u(t), t), t))*t^2+57*(diff(u(t), t))*t-105*u(t))/t^3

(4.2)

Its Hankel transform is a simple algebraic expression

>

(4.3)

An example with formula_plus

>

((diff(diff(diff(diff(u(t), t), t), t), t))*t^4+38*(diff(diff(diff(u(t), t), t), t))*t^3+477*(diff(diff(u(t), t), t))*t^2+2295*(diff(u(t), t))*t+3465*u(t))/t^4

(4.4)

>

(4.5)

In the case of hankel , not just differential operators but also several new transforms are now computable

>

piecewise(nu = 0, Dirac(k)/k, nu/k^2)

(4.6)

>

piecewise(And(nu = 0, m = 0), Dirac(k)/k, 2^(m+1)*k^(-m-2)*GAMMA(1+(1/2)*m+(1/2)*nu)/GAMMA((1/2)*nu-(1/2)*m))

(4.7)

>

Mellin and Hankel transform solutions for Partial Differential Equations with Boundary Conditions

In previous Maple releases, the Fourier and Laplace transforms were used to compute exact solutions to PDE problems with boundary conditions. Now, Mellin and Hankel transforms are also used for that same purpose.

Example:

>

pde := x^2*(diff(u(x, y), x, x))+x*(diff(u(x, y), x))+diff(u(x, y), y, y) = 0

iv := u(x, 0) = 0, u(x, 1) = piecewise(0 <= x and x < 1, 1, 1 < x, 0)

>

(5.1)

As usual, you can let pdsolve choose the solving method, or indicate the method yourself:

>

pde := diff(u(r, t), r, r)+(diff(u(r, t), r))/r+diff(u(r, t), t, t) = -Q__0*q(r)
iv := u(r, 0) = 0

>

u(r, t) = Q__0*(-hankel(exp(-s*t)*hankel(q(r), r, s, 0)/s^2, s, r, 0)+hankel(hankel(q(r), r, s, 0)/s^2, s, r, 0))

(5.2)

It is sometimes preferable to see these solutions in terms of more familiar integrals. For that purpose, use

>

u(r, t) = Q__0*(-(Int(exp(-s*t)*(Int(q(r)*r*BesselJ(0, r*s), r = 0 .. infinity))*BesselJ(0, r*s)/s, s = 0 .. infinity))+Int((Int(q(r)*r*BesselJ(0, r*s), r = 0 .. infinity))*BesselJ(0, r*s)/s, s = 0 .. infinity))

(5.3)

An example where the hankel transform is computable to the end:

>

pde := c^2*(diff(u(r, t), r, r)+(diff(u(r, t), r))/r) = diff(u(r, t), t, t)
iv := u(r, 0) = A*a/(a^2+r^2)^(1/2), (D[2](u))(r, 0) = 0

>

u(r, t) = (1/2)*A*a*((-c^2*t^2+(2*I)*a*c*t+a^2+r^2)^(1/2)+(-c^2*t^2-(2*I)*a*c*t+a^2+r^2)^(1/2))/((-c^2*t^2-(2*I)*a*c*t+a^2+r^2)^(1/2)*(-c^2*t^2+(2*I)*a*c*t+a^2+r^2)^(1/2))

(5.4)

>

Download Integral_Transforms_(revamped).mw

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

The Physics Updates, how to install in M...

Posted: ecterrab 14540 Product: Maple 2019

March 17 2019

5 4

Hi

The Physics Updates for Maple 2019 (current v.331 or higher) is already available for installation via MapleCloud. This version contains further improvements to the Maple 2019 capabilities for solving PDE & BC as well as to the tensor simplifier. To install these Updates,

Open Maple,
Click the MapleCloud icon in the upper-right corner to open the MapleCloud toolbar
In the MapleCloud toolbar, open Packages
Find the Physics Updates package and click the install button, it is the last one under Actions
To check for new versions of Physics Updates, click the MapleCloud icon. If the Updates icon has a red dot, click it to install the new version

Note that the first time you install the Updates in Maple 2019 you need to install them from Packages, even if in your copy of Maple 2018 you had already installed these Updates.

Also, at this moment you cannot use the MapleCloud to install the Physics Updates for Maple 2018. So, to install the last version of the Updates for Maple 2018, open Maple 2018 and enter PackageTools:-Install("5137472255164416", version = 329, overwrite)

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

A Complete Guide for Tensor computation...

Posted: ecterrab 14540 Product: Maple

January 23 2019

7 1

A Complete Guide for performing Tensors computations using Physics

This is an old request, a complete guide for using Physics to perform tensor computations. This guide, shown below with Sections closed, is linked at the end of this post as a pdf file with all the sections open, and also as a Maple worksheet that allows for reproducing its contents. Most of the computations shown are reproducible in Maple 2018.2.1, and a significant part also in previous releases, but to reproduce everything you need to have the Maplesoft Physics Updates version 283 or higher installed. Feedback one how to improve this presentation is welcome.

Physics is a package developed by Maplesoft, an integral part of the Maple system. In addition to its commands for Quantum Mechanics, Classical Field Theory and General Relativity, Physics includes 5 other subpackages, three of them also related to General Relativity: Tetrads , ThreePlusOne and NumericalRelativity (work in progress), plus one to compute with Vectors and another related to the Standard Model (this one too work in progress).

The presentation is organized as follows. Section I is complete regarding the functionality provided with the Physics package for computing with tensors in Classical and Quantum Mechanics (so including Euclidean spaces), Electrodynamics and Special Relativity. The material of section I is also relevant in General Relativity, for which section II is all devoted to curved spacetimes. (The sub-section on the Newman-Penrose formalism needs to be filled with more material and a new section devoted to the EnergyMomentum tensor is appropriate. I will complete these two things as time permits.) Section III is about transformations of coordinates, relevant in general.

For an alphabetical list of the Physics commands with a brief one-line description and a link to the corresponding help page see Physics: Brief description of each command .

I. Spacetime and tensors in Physics

This section contains all what is necessary for working with tensors in Classical and Quantum Mechanics, Electrodynamics and Special Relativity. This material is also relevant for computing with tensors in General Relativity, for which there is a dedicated Section II. Curved spacetimes .

	Default metric and signature, coordinate systems

Tensors, their definition, symmetries and operations

Physics comes with a set of predefined tensors, mainly the spacetime metric , the space metric , and all the standard tensors of General Relativity. In addition, one of the strengths of Physics is that you can define tensors, in natural ways, by indicating a matrix or array with its components, or indicating any generic tensorial expression involving other tensors.

In Maple, tensor indices are letters, as when computing with paper and pencil, lowercase or upper case, latin or greek, entered using indexation, as in , and are displayed as subscripts as in . Contravariant indices are entered preceding the letter with ~, as in , and are displayed as superscripts as in . You can work with two or more kinds of indices at the same time, e.g., spacetime and space indices.

To input greek letters, you can spell them, as in mu for , or simpler: use the shortcuts for entering Greek characters . Right-click your input and choose Convert To → 2-D Math input to give to your input spelled tensorial expression a textbook high quality typesetting.

Not every indexed object or function is, however, automatically a tensor. You first need to define it as such using the Define command. You can do that in two ways:

1.	Passing the tensor being defined, say , possibly indicating symmetries and/or antisymmetries for its indices.

2.	Passing a tensorial equation where the left-hand side is the tensor being defined as in 1. and the right-hand side is a tensorial expression - or an Array or Matrix - such that the components of the tensor being defined are equal to the components of the tensorial expression.

After defining a tensor - say or - you can perform the following operations on algebraic expressions involving them

•	Automatic formatting of repeated indices, one covariant the other contravariant

•	Automatic handling of collisions of repeated indices in products of tensors

•	Simplify products using Einstein's sum rule for repeated indices.

•	SumOverRepeatedIndices of the tensorial expression.

•	Use TensorArray to compute the expression's components

•	TransformCoordinates .

If you define a tensor using a tensorial equation, in addition to the items above you can:

•	Get each tensor component by indexing, say as in or

•	Get all the covariant and contravariant components by respectively using the shortcut notation and .

•	Use any of the special indexing keywords valid for the pre-defined tensors of Physics; they are: definition, nonzero, and in the case of tensors of 2 indices also trace, and determinant.

•	No need to specify the tensor dependency for differentiation purposes - it is inferred automatically from its definition.

•	Redefine any particular tensor component using Library:-RedefineTensorComponent

•	Minimizing the number of independent tensor components using Library:-MinimizeTensorComponent

•	Compute the number of independent tensor components - relevant for tensors with several indices and different symmetries - using Library:-NumberOfTensorComponents .

The first two sections illustrate these two ways of defining a tensor and the features described. The next sections present the existing functionality of the Physics package to compute with tensors.

	Defining a tensor passing the tensor itself

	Defining a tensor passing a tensorial equation

	Automatic formatting of repeated tensor indices and handling of their collisions in products

	Tensor symmetries

	Substituting tensors and tensor indices

	Simplifying tensorial expressions

	SumOverRepeatedIndices

	Visualizing tensor components - Library:-TensorComponents and TensorArray

	Modifying tensor components - Library:-RedefineTensorComponent

	Enhancing the display of tensorial expressions involving tensor functions and derivatives using CompactDisplay

	The LeviCivita tensor and KroneckerDelta

	The 3D space metric and decomposing 4D tensors into their 3D space part and the rest

	Total differentials, the d_[mu] and dAlembertian operators

	Tensorial differential operators in algebraic expressions

	Inert tensors

	Functional differentiation of tensorial expressions with respect to tensor functions

	The Pauli matrices and the spacetime Psigma[mu] 4-vector

	The Dirac matrices and the spacetime Dgamma[mu] 4-vector

	Quantum not-commutative operators using tensor notation

II. Curved spacetimes

Physics comes with a set of predefined tensors, mainly the spacetime metric , the space metric , and all the standard tensors of general relativity, respectively entered and displayed as: Einstein[mu,nu] = , Ricci[mu,nu] = , Riemann[alpha, beta, mu, nu] = , Weyl[alpha, beta, mu, nu], = , and the Christoffel symbols Christoffel[alpha, mu, nu] = and Christoffel[~alpha, mu, nu] = respectively of first and second kinds. The Tetrads and ThreePlusOne subpackages have other predefined related tensors. This section is thus all about computing with tensors in General Relativity.

	Loading metrics from the database of solutions to Einstein's equations

	Setting the spacetime metric indicating the line element or a Matrix

	Covariant differentiation: the D_[mu] operator and the Christoffel symbols

	The Einstein, Ricci, Riemann and Weyl tensors of General Relativity

	A conversion network for the tensors of General Relativity

	Tetrads and the local system of references - the Newman-Penrose formalism

	The ThreePlusOne package and the 3+1 splitting of Einstein's equations

	III. Transformations of coordinates

	See Also
	Physics , Conventions used in the Physics package , Physics examples , Physics Updates

Download Tensors_-_A_Complete_Guide.mw, or the pdf version with sections open: Tensors_-_A_Complete_Guide.pdf

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

Overview of the Physics Updates...

Posted: ecterrab 14540 Product: Maple

January 11 2019

8 0

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.	Tensor product of Quantum States using Dirac's Bra-Ket Notation

b.	Coherent States in Quantum Mechanics

c.	The Zassenhaus formula and the algebra of the Pauli matrices

d.	Multivariable Taylor series of expressions involving anticommutative (Grassmannian) variables

e.	New SortProducts command

f.	A Complete Guide for Tensor computations using Physics

•	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.	The Physics project at Maplesoft

2.	Mini-Course: Computer Algebra for Physicists

3.	A Complete Guide for Tensor computations using Physics

4.	Perimeter Institute-2015, Computer Algebra in Theoretical Physics (I)

5.	IOP-2016, Computer Algebra in Theoretical Physics (II)

6.	ACA-2017, Computer Algebra in Theoretical Physics (III)

•	General Relativity

7.	General Relativity using Computer Algebra

8.	Exact solutions to Einstein's equations

9.	Classification of solutions to Einstein's equations and the ThreePlusOne (3 + 1) package

10.	Tetrads and Weyl scalars in canonical form

11.	Equivalence problem in General Relativity

12.	Automatic handling of collision of tensor indices in products

13.	Minimize the number of tensor components according to its symmetries

•	Quantum Mechanics

14.	Quantum Commutation Rules Basics

15.	Quantum Mechanics: Schrödinger vs Heisenberg picture

16.	Quantization of the Lorentz Force

17.	Magnetic traps in cold-atom physics

18.	The hidden SO(4) symmetry of the hydrogen atom

19.	(I) Ground state of a quantum system of identical boson particles

20.	(II) The Gross-Pitaevskii equation and Bogoliubov spectrum

21.	(III) The Landau criterion for Superfluidity

22.	Simplification of products of Dirac matrices

23.	Algebra of Dirac matrices with an identity matrix on the right-hand side

24.	Factorization with non-commutative variables

25.	Tensor Products of Quantum State Spaces

26.	Coherent States in Quantum Mechanics

27.	The Zassenhaus formula and the Pauli matrices

•	Physics package generic functionality

28.	Automatic simplification and a new Assume (as in "extended assuming")

29.	Wirtinger derivatives and multi-index summation

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

Download Physics-Updates.mw

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

The Zassenhaus formula and the Pauli mat...

Posted: ecterrab 14540 Product: Maple

January 11 2019

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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

$"0=((∑)(lambda^n)/(n!) {A^n,B})+2 lambda ((∑) (∑)(lambda^(n+m))/(n! m!) {A^m,B^n,C[2]})+3 lambda^2 ((∑) (∑) (∑)(lambda^(n+m+k))/(n! m! k!) {A^k,B^m,(C[2])^n,C[3]})+ ..."$

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

${B, A^0} = B, {B, A^(n+1)} = %Commutator(A, {A^n, B^n})$

${C[j], B^n, A^0} = {C[j], B^n}, {C[j], A^m, B^n} = %Commutator(A, {A^`-`(m, 1), B^n, C[j]^k})$

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

$"0=((∑)(lambda^n)/(n!) {A^n,B})+2 lambda ((∑) (∑)(lambda^(n+m))/(n! m!) {A^m,B^n,C[2]})+3 lambda^2 ((∑) (∑) (∑)(lambda^(n+m+k))/(n! m! k!) {A^k,B^m,(C[2])^n,C[3]})+..."$

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:

$0 = Sum(lambda^n*{B, A^n}/factorial(n), n = 1 .. 2)+2*lambda*(Sum(Sum(lambda^(n+m)*{C[2], A^m, B^n}/(factorial(n)*factorial(m)), n = 0 .. 1), m = 0 .. 1))+3*lambda^2*(Sum(Sum(Sum(lambda^(n+m+k)*{C[3], A^k, B^m, C[2]^n}/(factorial(n)*factorial(m)*factorial(k)), n = 0 .. 0), m = 0 .. 0), k = 0 .. 0))$

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

$0 = Sum(lambda^n*{B, A^n}/factorial(n), n = 1)+2*lambda*(Sum(lambda^(n+m)*{C[2], A^m, B^n}/(factorial(n)*factorial(m)), n+m = 1))+3*lambda^2*(Sum(lambda^(n+m+k)*{C[3], A^k, B^m, C[2]^n}/(factorial(n)*factorial(m)*factorial(k)), n+m+k = 0))$

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

${B, A^0} = B, {B, A^(n+1)} = %Commutator(A, {A^n, B^n})$

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

>

H := sum(lambda^n*F(A, B, n)/factorial(n), n = 2)+2*lambda*(sum(lambda^(n+m)*F(A, F(B, C[2], n), m)/(factorial(n)*factorial(m)), n+m = 1))+3*lambda^2*(sum(lambda^(n+m+k)*F(A, F(B, F(C[2], C[3], n), m), k)/(factorial(n)*factorial(m)*factorial(k)), n+m+k = 0))

(1/2)*lambda^2*%Commutator(Physics:-Psigma[1], %Commutator(Physics:-Psigma[1], Physics:-Psigma[3]))+2*lambda*(lambda*%Commutator(Physics:-Psigma[1], I*Physics:-Psigma[2])+lambda*%Commutator(Physics:-Psigma[3], I*Physics:-Psigma[2]))+3*lambda^2*C[3]

(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,

>

C[3] = (2/3)*Physics:-Psigma[3]-(4/3)*Physics:-Psigma[1]

(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:

>

H := unapply(sum(lambda^n*F(A, B, n)/factorial(n), n = N)+2*lambda*(sum(lambda^(n+m)*F(A, F(B, C[2], n), m)/(factorial(n)*factorial(m)), n+m = N-1))+3*lambda^2*(sum(lambda^(n+m+k)*F(A, F(B, F(C[2], C[3], n), m), k)/(factorial(n)*factorial(m)*factorial(k)), n+m+k = N-2)), N)

(13)

Now we have

>

(14)

>

(15)

The following is already equal to (11)

>

(1/2)*lambda^2*%Commutator(Physics:-Psigma[1], %Commutator(Physics:-Psigma[1], Physics:-Psigma[3]))+2*lambda*(lambda*%Commutator(Physics:-Psigma[1], I*Physics:-Psigma[2])+lambda*%Commutator(Physics:-Psigma[3], I*Physics:-Psigma[2]))+3*lambda^2*C[3]

(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

>

for j to 4 do C[j+1] := Simplify(solve(H(j), C[j+1])) end do

(2/3)*Physics:-Psigma[3]-(4/3)*Physics:-Psigma[1]

-(8/9)*Physics:-Psigma[1]-(158/45)*Physics:-Psigma[3]-((16/3)*I)*Physics:-Psigma[2]

(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

>

sum(lambda^n*F(Physics:-Psigma[1], Physics:-Psigma[3], n)/factorial(n), n = N)+2*lambda*(sum(lambda^(n+m)*F(Physics:-Psigma[1], F(Physics:-Psigma[3], C[2], n), m)/(factorial(n)*factorial(m)), n+m = N-1))

(20)

>

sum(lambda^n*F(Physics:-Psigma[1], Physics:-Psigma[3], n)/factorial(n), n = N)+2*lambda*(sum(lambda^(n+m)*F(Physics:-Psigma[1], F(Physics:-Psigma[3], C[2], n), m)/(factorial(n)*factorial(m)), n+m = N-1))+3*lambda^2*(sum(lambda^(n+m+k)*F(Physics:-Psigma[1], F(Physics:-Psigma[3], F(C[2], C[3], n), m), k)/(factorial(n)*factorial(m)*factorial(k)), n+m+k = N-2))

(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

>

for j to 9 do C[j+1] := Simplify(solve(H(j), C[j+1])) end do

(2/3)*Physics:-Psigma[3]-(4/3)*Physics:-Psigma[1]

-(8/9)*Physics:-Psigma[1]-(158/45)*Physics:-Psigma[3]-((16/3)*I)*Physics:-Psigma[2]

(1030/81)*Physics:-Psigma[1]-(8/81)*Physics:-Psigma[3]+((1078/405)*I)*Physics:-Psigma[2]

((11792/243)*I)*Physics:-Psigma[2]+(358576/42525)*Physics:-Psigma[1]+(12952/135)*Physics:-Psigma[3]

(87277417/492075)*Physics:-Psigma[1]+(833718196/820125)*Physics:-Psigma[3]+((35837299048/17222625)*I)*Physics:-Psigma[2]

-((449018539801088/104627446875)*I)*Physics:-Psigma[2]-(263697596812424/996451875)*Physics:-Psigma[1]+(84178036928794306/2197176384375)*Physics:-Psigma[3]

(3226624781090887605597040906/21022858292748046875)*Physics:-Psigma[1]+(200495118165066770268119656/200217698026171875)*Physics:-Psigma[3]+((2185211616689851230363020476/4204571658549609375)*I)*Physics:-Psigma[2]

(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

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