ecterrab

Computer Algebra for Theoretical Physics...

Posted: ecterrab 13431

October 15 2014

12 4

Last week the Physics package was presented in a talk at the Perimeter Institute for Theoretical Physics and in a combined Applied Mathematics and Physics Seminar at the University of Waterloo. The presentation at the Perimeter Institute got recorded. It was a nice opportunity to surprise people with the recent advances in the package. It follows the presentation with sections closed, and at the end there is a link to a pdf with the sections open and to the related worksheet, used to run the computations in real time during the presentation.

COMPUTER ALGEBRA FOR THEORETICAL PHYSICS

Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system implemented most of the mathematical objects and mathematics used in theoretical physics computations, and dramatically approximated the notation used in the computer to the one used in paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum. In connection, in this talk the Physics project at Maplesoft is presented and the resulting Physics package illustrated tackling problems in classical and quantum mechanics, general relativity and field theory. In addition to the 10 a.m lecture, there will be a hands-on workshop at 1pm in the Alice Room.

... Why computers?

We can concentrate more on the ideas instead of on the algebraic manipulations

We can extend results with ease

We can explore the mathematics surrounding a problem

We can share results in a reproducible way

Representation issues that were preventing the use of computer algebra in Physics

Notation and related mathematical methods that were missing:

coordinate free representations for vectors and vectorial differential operators,

covariant tensors distinguished from contravariant tensors,

functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices

Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics

Inert representations of operations, mathematical functions, and related typesetting were missing:

inert versus active representations for mathematical operations

ability to move from inert to active representations of computations and viceversa as necessary

hand-like style for entering computations and texbook-like notation for displaying results

Key elements of the computational domain of theoretical physics were missing:

ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions

ability to perform computations taking into account custom-defined algebra rules of different kinds

(problem related commutator, anticommutator, bracket, etc. rules)

	Vector and tensor notation in mechanics, electrodynamics and relativity

	Dirac's notation in quantum mechanics

•

Computer algebra systems were not originally designed to work with this compact notation, having attached so dense mathematical contents, active and inert representations of operations, not commutative and customizable algebraic computational domain, and the related mathematical methods, all this typically present in computations in theoretical physics.

•	This situation has changed. The notation and related mathematical methods are now implemented.

Tackling examples with the Physics package

Classical Mechanics

Inertia tensor for a triatomic molecule

Problem: Determine the Inertia tensor of a triatomic molecule that has the form of an isosceles triangle with two masses in the extremes of the base and mass in the third vertex. The distance between the two masses is equal to a, and the height of the triangle is equal to h.

	Solution

Quantum mechanics

Quantization of the energy of a particle in a magnetic field

Show that the energy of a particle in a constant magnetic field oriented along the z axis can be written as

where and are creation and anihilation operators.

	Solution

	The quantum operator components of satisfy

Unitary Operators in Quantum Mechanics

(with Pascal Szriftgiser, from Laboratoire PhLAM, Université Lille 1, France)

A linear operator U is unitary if , in which case, Unitary operators are used to change the basis inside an Hilbert space, which physically means changing the point of view of the considered problem, but not the underlying physics. Examples: translations, rotations and the parity operator.

	1) Eigenvalues of an unitary operator and exponential of Hermitian operators

	2) Properties of unitary operators

	3) Schrödinger equation and unitary transform

	4) Translation operators

Classical Field Theory

The field equations for a quantum system of identical particles

Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE). This equation is particularly useful to describe Bose-Einstein condensates (BEC).

	Solution

	The field equations for the model

	Maxwell equations departing from the 4-dimensional Action for Electrodynamics

General Relativity

Given the spacetime metric,

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = -exp(lambda(r)), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = exp(nu(r))}))$

a) Compute the trace of

where is some function of the radial coordinate, is the Ricci tensor, is the covariant derivative operator and is the stress-energy tensor

$T[alpha, beta] = (Matrix(4, 4, {(1, 1) = 8*exp(lambda(r))*Pi, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 8*r^2*Pi, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 8*r^2*sin(theta)^2*Pi, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 8*exp(nu(r))*Pi*epsilon}))$

b) Compute the components of the traceless part of of item a)

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of obtained in b)

Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.

	a) The trace of

	b) The components of the traceless part of

	c) An exact solution for the nonlinear system of differential equations conformed by the components of

The Physics Project

"Physics" is a software project at Maplesoft that started in 2006. The idea is to develop a computational symbolic/numeric environment specifically for Physics, targeting educational and research needs in equal footing, and resembling as much as possible the flexible style of computations used with paper and pencil. The main reference for the project is the Landau and Lifshitz Course of Theoretical Physics.

A first version of "Physics" with basic functionality appeared in 2007. Since then the package has been growing every year, including now, among other things, a searcheable database of solutions to Einstein equations and a new dedicated programming language for Physics.

Since August/2013, weekly updates of the Physics package are distributed on the web, including the new developments related to our plan as well as related to people's feedback.

Presentation_at_PI_and_UW.pdf Presentation_at_PI_and_UW.mw

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

DEs and Mathematical Functions Updates 2...

Posted: ecterrab 13431

June 23 2014

3 13

Hi,
The FunctionAdvisor project is currently developing at full speed. During the last two months, a significant amount of new conversion routines and mathematical information for Jacobi elliptic and Jacobi Theta functions, on identities, periodicity, transformations, etc. got added to the conversion network for mathematical functions and to the FunctionAdvisor. The previous months was the turn of the set of complex components, added to the network. Developments regarding the simplification and integration of special functions (e.g SphericalY for computing spherical harmonics or Dirac), as well as fixes to the numerical evaluation of JacobiAM, `assuming` and to differential equation subroutines are also part of the update.

These developments are available to everybody as usual in the Maplesoft R&D Differential Equations and Mathematical Functions webpage. Below there is a list of the latest developments as seen in the worksheet that comes in the zip with the DEsAndMathematicalFunctions update.

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

Physics updates 2014 (Jan-Jun)...

Posted: ecterrab 13431

June 19 2014

8 0

Hi,
An interesting sequence of enhancements and new developments happened in the Physics package during this first half of the year. During the last month, improvements happened in the handling of Vectorial expressions and quantum mechanics using Dirac’s notation. During April and part of May it was the turn of general relativity enhancements.

Some of the developments are also interesting beyond Physics. For example: it is now possible to multiply equations. Suppose you have A = B (1), and C = D (2), multiplying as in (1) (2) now results in lhs((1)) lhs((2)) = rhs((1)) rhs((2)), saving a lot of typing. You can also perform (1)/(2) or (1)^2. Some enhancements in Physics related simplification, integration, `assuming`, and typesetting - e.g. the simplification and integration of spherical harmonics (SphericalY function) are also part of the update.

These developments are available to everybody as usual in the Maplesoft R&D Physics webpage. Below there is a list of the developments for the last month as seen in the worksheet that comes in the zip with the Physics update.

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

DEs and Mathematical Functions Updates...

Posted: ecterrab 13431

May 06 2014

2 0

This is the first presentation of updates for the DE and Mathematical Functions programs of Maple 18. It includes several improvements, all in the Mathematical Functions sector, as well as some fixes. The update and instructions for its installation are available on the Maplesoft R&D webpage for DEs and mathematical functions. Some of the items below were mentioned here in Mapleprimes - you are welcome to present suggestions or issues; if possible they will be addressed right away in the next update.

Filling gaps in the FunctionAdvisor regarding all the 6 complex components: abs, argument, conjugate, Im, Re, signum, as well as regarding Heaviside (step function), Dirac, min and max.
Fix the simplification and differentation rule for doublefactorial
Make convert(..., hypergeometric) work the same way as convert(blabla, hypergeom)
Implement integral forms for Heaviside(z) and JacobiAM(z, k) via convert(..., Int)
Implement appropriate display for the inert %intat function as well as its conversion to the inert Int
Make the FunctionAdvisor/DE return not just the PDE system satisfied by f(z, k) = JacobiAM(z, k)and also (new) the ODE satisfied by f(z) = JacobiAM(z, k)
Fix conversion rule from Heaviside(z) to Sum
Fix unexpected error interruption when differentiating min(...) and max(...) containing more than three arguments
Fix issue in simplify/conjugate
Improvement in expand/int: factors in disguise are put outside the integration sign
Various improvements in the case of multiple integrals involving the Dirac function
Make Intat fully inert (before it was evaluating its arguments)
Make value of inert indexed objects work

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

Superfluidity in Quantum Mechanics...

Posted: ecterrab 13431

April 13 2014

7 0

The attached presentation is the last one of a sequence of three on Quantum Mechanics using Computer Algebra, covering the field equation for a quantum system of identical particles, its stationary solutions and the equations for small perturbations around them and, in this third presentation, the conditions for superfluidity of such a system of identical particles at low temperature. The novelty is again in how to tackle these problems in a computer algebra worksheet.

The Landau criterion for Superfluidity

Pascal Szriftgiser¹ and Edgardo S. Cheb-Terrab²

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

(2) Maplesoft, Canada

A Bose-Einstein Condensate (BEC) is a medium constituted by identical bosonic particles at very low temperature that all share the same quantum wave function. Let's consider an impurity of mass M, moving inside a BEC, its interaction with the condensate being weak. At some point the impurity might create an excitation of energy and momentum . We assume that this excitation is well described by Bogoliubov's equations for small perturbations around the stationary solutions of the field equations for the system. In that case, the Landau criterion for superfluidity states that if the impurity velocity is lower than a critical velocity (equal to the BEC sound velocity), no excitation can be created (or destroyed) by the impurity. Otherwise, it would violate conservation of energy and momentum. So that, if < the impurity will move within the condensate without dissipation or momentum exchange, the condensate is superfluid (Phys. Rev. Lett. 85, 483 (2000)). Note: low temperature liquid ⁴He is a well known example of superfluid that can, for instance, flow through narrow capillaries with no dissipation. However, for superfluid helium, the critical velocity is lower than the sound velocity. This is explained by the fact that liquid ⁴He is a strongly interacting medium. We are here rather considering the case of weakly interacting cold atomic gases.

Landau criterion for superfluidity

Background: For a BEC close to its ground state (at temperature T = 0 K), its excitations are well described by small perturbations around the stationary state of the BEC. The energy of an excitation is then given by the Bogoliubov dispersion relation (derived previously in Mapleprimes "Quantum Mechanics using computer algebra II").

epsilon[k] = `&hbar;`*omega[k] and `&hbar;`*omega[k] = `&+-`(sqrt(k^4*`&hbar;`^4/(4*m^2)+k^2*`&hbar;`^2*G*n/m))

where G is the atom-atom interaction constant, n is the density of particles, m is the mass of the condensed particles, k is the wave-vector of the excitations and their pulsation ( time the frequency). Typically, there are two possible types of excitations, depending on the wave-vector :

•	In the limit: , with , this relation is linear in and is typical of a massless quasi-particle, i.e. a phonon excitation.

•	In the limit: , which is the dispersion relation of a free particle of mass i.e. one single atom of the BEC.

Problem: An impurity of mass moves with velocity within such a condensate and creates an excitation with wave-vector . After the interaction process, the impurity is scattered with velocity .

a) Departing from Bogoliubov's dispersion relation, plus energy and momentum conservation, show that, in order to create an excitation, the impurity must move with an initial velocity

When , no excitation can be created and the impurity moves through the medium without dissipation, as if the viscosity is 0, characterizing a superfluid. This is the Landau criterion for superfluidity.

b) Show that when the atom-atom interaction constant (repulsive interactions), this value is equal to the group velocity of the excitation (speed of sound in a condensate).