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I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This third presentation, about "Computer Algebra in Theoretical Physics", describes the Physics project at Maplesoft, also my first research project at University, that evolved into the now well-known Maple Physics package. This is a unique piece of software and perhaps the project I most enjoy working.

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

 

 

Computer Algebra in Theoretical Physics

 

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Abstract:

 

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 have implemented most of the mathematical objects and mathematics used in theoretical physics computations, and have dramatically approximated the notation used in the computer to the one used with paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum.

 

In this talk, the Physics project at Maplesoft is presented and the resulting Physics package is illustrated by tackling problems in classical and quantum mechanics, using tensor and Dirac's Bra-Ket notation, general relativity, including the equivalence problem, and classical field theory, deriving field equations using variational principles.

 

 

 

 

... and why computer algebra?

 

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

   

Classical Mechanics

 

*Inertia tensor for a triatomic molecule

   

Quantum mechanics

 

*The quantum operator components of  `#mover(mi("L",mathcolor = "olive"),mo("→",fontstyle = "italic"))` satisfy "[L[j],L[k]][-]=i `ε`[j,k,m] L[m]"

   

*Unitary Operators in Quantum Mechanics

 

*Eigenvalues of an unitary operator and exponential of Hermitian operators

   

*Properties of unitary operators

 

 

Consider two set of kets " | a[n] >" and "| b[n] >", each of them constituting a complete orthonormal basis of the same space.

*Verify that "U=(&sum;) | b[k] >< a[k] |" , maps one basis to the other, i.e.: "| b[n] >=U | a[n] >"

   

*Show that "U=(&sum;) | b[k] > < a[k] | "is unitary

   

*Show that the matrix elements of U in the "| a[n] >" and  "| b[n] >" basis are equal

   

Show that A and `&Ascr;` = U*A*`#msup(mi("U"),mo("&dagger;"))`have the same spectrum (eigenvalues)

   

Schrödinger equation and unitary transform

   

Translation operators using Dirac notation

   

*Quantization of the energy of a particle in a magnetic field

   

Classical Field Theory

 

The field equations for the lambda*Phi^4 model

   

*Maxwell equations departing from the 4-dimensional Action for Electrodynamics

   

*The Gross-Pitaevskii field equations for a quantum system of identical particles

   

General Relativity

 

Exact Solutions to Einstein's Equations  Lambda*g[mu, nu]+G[mu, nu] = 8*Pi*T[mu, nu]

   

*"Physical Review D" 87, 044053 (2013)

   

The Equivalence problem between two metrics

   

*On the 3+1 split of the 4D Einstein equations

   

Tetrads and Weyl scalars in canonical form

   

 

 


 

Download Physics.mw

Download Physics.pdf

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

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