Next term, I'm teaching our sophomore-level "Modern Physics" class again. "Modern Physics," in ecuation terms, really means "Early 20th Century Physics"-- it's a couple of weeks of Special Relativity, followed by several weeks of basic Quantum Mechanics, with a mad 2-3 week sprint at the end where I try to cover as much material as possible relating to applications of Relativity and QM.
One of the things I did last year with this class was to try to introduce a little computational work using Mathematica. It's a useful tool for our students to learn about, and it allows you to discuss some problems that can only be done numerically. As with most things that I try for the first time, it was ok, but not great, and I think I really need to spend more time introducing the students to Mathematica, and integrating it into the class a little more thoroughly.
I've got a fair number of examples of quantum-mechanical problems that can be solved using Mathematica, so I'm probably ok there, but the problem is those first two weeks. I'd really like to get them going early, but I don't have many ideas for ways to use Mathematica that are relevant to Special Relativity. There are a few really basic things that will serve as a good introduction to the program-- have them learn how to define functions and make graphs by looking at the Lorentz factor γ as a function of velocity, and that sort of thing-- but nothing beyond that really leaps to mind.
But, this is one of the nice things about having a blog read by so many really smart people who are susceptible to flattery: I can ask my readers. So, really smart readers who are susceptible to flattery, any suggestions of things to study about Special Relativity that would make good Mathematica exercises?
Do you teach them world-line diagrams? If so, you could make them draw world-line diagrams for things like the pole-and-barn paradox, and Lorentz transform them between frames. You'd probably have to do a bit of work in advance to help the students with the transformations, though...
I wrote myself a windows Lorentz transform equation for my own modern physics course, to show them the paradoxes and the transformations 'real-time'. The results were somewhat mixed, though, because I didn't make it easy enough to use for a lecture setting.
Send an e-mail to Mathematica support or Wolfram himself. They no doubt have past applications catalogued and are always eager to add new tricks.
A quick lesson on setting up a Runge-Kutta routine to solve a simple ODE (say, Burgers) will teach the basics of numerical programming and Mathematica.
I don't know how in depth you are going to get in relativity, but you could plot the electric field of a point charge in the rest frame, and then transform into a frame where the charge is moving at some sizable chunk of c.
The distribution of the electric field is no longer isotropic,IIRC the field lines are more dense in the direction orthogonal to the direction of motion.
Also, you have magnetic fields which you can plot on top of the electric field.
The problem is that in order to do justice to relatavistic electromagnetism you have to spend a fair bit of time on the nitty gritty tensor stuff, which was certainly beyond the scope of my modern physics course, and probably beyond the scope of this one.
You might be able to explain it in terms of a sphere of constant potential which gets squashed due to length contraction. This analogy is off the top of my head, so I don't know if it is a valid way of thinking of the situation.
Another thing you could do is have Mathematica generate a few hundred random points in 3 dimensions (x_i,y_i,z_i) such that x_i>0. Then set the ViewPoint variable to look at the points on the origin facing in the +x direction. Pretend these are stars
You could then length contract the x component of each star so that all your stars move towards the edges of your graph. I am not sure how well this would work, and it very well might be a huge pain to do.
I don't think world line diagrams would work that well, they aren't very hard to draw by hand, and are sort of a pain to draw on the computer, especially if bits are unknown. I remember a rather tricky problem my prof gave which involved a relativistic shootout between Klingon and Romulan ships where you had to draw events at "some time later". Which would be sort of finnicky to get right on the computer.
Do three spatial dimensions.
Generally, talking about special relativity, we set y=z=0 for everything, and worry only about x and t. And, with good reason! The vast majority of the interesting effects show up moving in one spatial direction. Plus, it makes the math tractable.
However, you can give them the equations (which are just adding in rotations that soph. Physics majors who did any Tau=I*omega stuff in their frosh year can handle). It makes the algebra very painful without appreciable physical insight, so it's not appropriate for written homeowrk. (Save that kind of stuff for their Classical Mechanics and E&M classes... no, just kidding, although too often it seems that way.)
One thing you can work out is the "effective rotaton" you see from objects. You can do it in 1-d thinking about angles and speed of light stuff, but perhaps you could assign the problem to do it with mathematica, and then ask them to analyze it later to see if it all made sense.
You can also do various accelerating particles; messed up acceleration, if you're willing to go numerical.
I don't usually talk about world line diagrams, partly because we only have about two weeks to spend on relativity, but mostly because I don't find the wilted-axis diagrams terribly illuminating. For simple kinematics problems, it tends to just make things more confusing.
We do talk about transforms in 3-D, though we generally don't do much off the x-axis. I do tend to throw on one exam question asking about the velocity of something shot out to the side in a moving frame, so that might be a good way to go.
Maybe I'll try to work up some 3-D scenarios-- plot the observed dimensions of a sphere accelerating along one axis, and that sort of thing.
You can find various examples online at the Mathematica website for various fields (not sure if these are mainly user submitted). For example, relativity notebooks can be found here:
No Mathematica hints, but look at Craig Savage's website for some cool movies; also Wheeler and Taylor's book has some great paradoxes, like one with a U shaped piece and a T shaped piece, moving relative to one another. You put a bomb on one piece. If the T shrinks, no bomb. If the U shrinks, BOMB! Now something is off here, cause either the bomb goes off or it doesn't. I'm giving you a very short version here, but its a cool thing, and Taylor and Wheeler is full of good stuff.
as far as E&M, you'd be almost negligent to not mention that a stationary charge makes E, and if you are moving by at a constant velocity, you see E AND B cause you see a current. Why did Lorentz get his name on the equations if Big Al did this? Lorentz showed that Maxwells eqns were invariant under the L transformation, classical mechanics was invariant under Galilean. Big Al thought there should be only one rule for going to moving systems, Galilean didn't work for E&M, Lorentz didn't work for F=ma, one of them had to give! Now the transformations for E and B are messy, but A and Phi form just one more 4 vector, like E and P, and X and T! Thats a hard course to teach, to do some relativity, basic QM, then sprint through chapters on atomic physics, molecules, solids, particles, etc...
Best relativity simulation: Cosmic rays.
Two points come to mind. First, assume isotropic random direction collisions (i.e. pool balls) in the rest frame of the primary particle and whatever it runs into, and use mathematica to show that these become extremely focused in the direction of motion in the earth's frame of reference.
Second, look at the shower growth. Show that the incoming shower of particles grow exponentially and arrive at the same time on the earth. Thus AGASA uses an RC time constant circuit for energy measurement of UHECRs. Charge is dumped into the ciruit when the shower arrives, and the time for the RC to drop gives the logarithm of the energy.
That should be enough. If not, for extra credit, show that if the primary particle is a tachyon, the cosmic ray shower will be extended in time and this will spoof AGASA's energy measurement to think that there was more energy than actually present. From the difference between AGASA and Fly's Eye energy estimates, compute a minimum survival time for tachyons in the atmosphere. (See astro-ph/0506166 )
Check out some of the packages at
There is a great animation that shows how time dilation affects the rate of pion decay.
Hey there, you may want to have a look at the Mathematica mailing list at http://smc.vnet.net/mathgroup.html, if you cannot find what you need there, just ask. MathGroup has proven to me to be very useful.
Check: "A Physicist's Guide to Mathematica"; by Patrick Tam. Page 299, example 3.4.5
(For those who might claim that the above example is just fudging observational representations... no duh. That's the point, isn't it? Out of the box, Mathematica isn't going to change axial origins in the middle of an operation just to suit the desire to prop up bygone physics icons to keep undergrad students interested. If they can't handle experimental uncertainty, send them over to a symbolic logic class instead (heh).