Introduction to Computation in Physics

Some places/ universities are really concerned about computer in physics education. APS-2004 W38

Using computers can be by machine coding (Assembly), high-level language coding (Fortran or Java …etc.) or mere running of simulations (e.g. PSPICE) and playing with physlet buttons as in a projectile motion physlet.

Please do not think that you will ‘learn’ Mathematica from this presentation though you will hopefully learn ‘about’ Mathematica.

What do I mean by: modeling, simulation, computation

A physicist!

Published his first scientific paper at the age of 15!!

Received his Ph.D. in theoretical physics from Caltech by the age of 20!!!

Worked at the Institute for Advanced Study in Princeton!!!!, then Professor of Physics, Mathematics, and Computer Science at the University of Illinois @ Urbana-Champaign!!!!!

President and CEO of Wolfram Research (!!!wow!!!)

Stephen Wolfram[the creator of Mathematica]

Is a (high level) programming language developed by Stephen Wolfram (first release in 1988) with special capabilities to do symbolic and numeric calculations in addition to graphing and many other features, as will be seen shortly.

Integrated environment for technical computing.

It has had a profound effect on the way computers are used in many technical and other fields.

First released in 1988; Current version:??8??

Used by, literally, millions worldwide.

Mathematica

If you do not know much Mathematica you should benefit from the electronic

help (the -electronic- Mathematica book)

Tour of featureshttp://www.wolfram.com/products/mathematica/tour/

Basic calculations, significant figures, trigonometry, complex notation

Algebra [systems of equations, eigensystem]

Graphics:

vector fields

3-D

parametric plotting

histograms

Symbolic computation

Numeric computation

Power of Mathematica

Special functions (Legendre, Bessel, Chebysef, …etc.)

Sound

Fitting data

Statistics

Communicate with external lists and Fortran.

Power of Mathematica

“The key intellectual

advance that made this possible was the invention of a new kind of

symbolic computer language that

could for the first time manipulate

the very wide range of objects

involved in technical

computing using only a fairly small number of basic

primitives.”Tour: Power of Mathematics

With Mathematica, the entire approach to problem solving can be drastically changed. We give some brief examples. ……………………………………………………………………………………DOUBLE PENDULUM: This is a topic that is generally treated as an "advanced" topic. With Mathematica, the solution is relatively straightforward. Once the solutions is obtained, the textbooks try to describe (in words) the general properties of the system, and the normal modes. (In particular, the property that the energy is transferred back and forth between the two segments of the pendulum.) With the animation capability of Mathematica, we do not need to lead the student to these conclusions, but we can point them in the general direction, and let them discover these results on their own by varying the amplitudes of the separate normal modes.……………………………………………………………………………………

HYDROGEN ATOM: In the standard solution of the hydrogen atom, the student is completely lost in the mathematics. Mathematica is able to recognize that the solution of the radial equation is a Laguerre polynomial, assemble the constants to form the principal quantum number, and plot the solutions. The student then has the energy and the curiosity to numerically investigate the behavior of the wavefunctions, and consider the disastrous consequences of choosing a non-integral value for the principal quantum number.

An excerpt from Mathematica for Physics, by Robert L. Zimmerman and Fredrick I. Olness

Projectile motion

Waves

Sound

How Mathematica benefits in phys-101/102?

By Dr. A. Al-Jalal using Mathematica

50 100 150 200 250 300

200

400

600

800

Classical Mechanics

Thermodynamics

Optics

Electronics

Quantum Mechanics

“Intermediate” level physics

-4 -2 0 2 4

-4

-2

0

2

4

-3 -2 -1 1 2 3

-6

-4

-2

2

4

6

Green's function

Canonical transformations

Chaos

Higher level physics

20 40 60 80 100

20

40

60

80

100

120

A series R,L,C electric circuit (assume R2 << 4L/C) initially carries no charge nor current. At time t = 0+ a volage V(t) is applied across the circuit such that:

V(t) = Vo e-t

Find the charge q(t) on the capacitor for t>0.

Hint: Use Green’s method; see M&T sect. 3.10

Computation and Mathematical Models

The concept is based on three elements:

1- The evolution of a system is referred to as the independent variable. Usually, this is the variable of time (t).2- The state variable is the finite dimensional vector variable {u1(t), u2(t), ….un(t)} deemed sufficient to describe the evolution of the physical state of the system. This is also called the dependent variable.3- The mathematical model of a system is an evolution equation suitable to define the evolution of the state variable {u} that is describing the system itself.

There are issues of validation, determinism and stochasticity that one needs to be concerned with! (c.f. see: Mechanics and Dynamical Systems with Mathematica, by Bellomo et. al.)

An Example: the setting sun “turning” red

1- The independent variable is time (t).

2- The state variable deemed sufficient to describe the evolution: color(t) and intensity(t).

3- The mathematical model of a system to define the time evolution of color and intensity: for light emitted, eye sensitivity and scattering in the atmosphere.

Check the code

Eric’s pages:http://scienceworld.wolfram.com/physics/

Web references that serve Mathematica

Info[rmation] Center:

• ~ five thousand Mathematica programs and document

• easy browsing and searchinghttp://library.wolfram.com/infocenter

Mechanics and Dynamical Systems with Mathematica:http://www.birkhauser.com/supplements/081764007X/Additional_Resources/index.html

The Mathematica Journal:http://www.mathematica-journal.com/issue/v9i1/

1) Mathematica for Calculus-Based Physics, by Marvin L. De Jong.

2) Mathematica for Physics, by Robert L. Zimmerman and Fredrick I. Olness.

3) Mechanics and Dynamical Systems with Mathematica, by Bellomo et. al.

4) Numerical and Analytical Methods for Scientists and Engineers, using Mathematica author: Daniel Dubin

5) Nonlinear Physics with Mathematica for Scientists and Engineers, by Richard H. Enns and George C. McGuire

6) Mathematical Methods Using Mathematica for Students of Physics and Related Fields, by Sadri Hassani

List of available books (some with CDs) for Mathematica in Physics

It would be wise to seriously study computational skills adequacy (or lack of) visa vis the current physics curriculum.

Conclusions:

I hope I have been able to get you more interested in computer programming for University pedagogy.

Mathematica is an interesting computer program that is very useful in physics.

Using Mathematica is fundamentally different from using simulations or playing with physlets.