I’m frightfully late on following up on this, but ElKement has another entry in the series regarding quantum field theory, this one engagingly titled “On The Relation Of Jurassic Park and Alien Jelly Flowing Through Hyperspace”. The objective is to introduce the concept of phase space, a way of looking at physics problems that marks maybe the biggest thing one really needs to understand if one wants to be not just a physics major (or, for many parts of the field, a mathematics major) and a grad student.
As an undergraduate, it’s easy to get all sorts of problems in which, to pick an example, one models a damped harmonic oscillator. A good example of this is how one models the way a car bounces up and down after it goes over a bump, when the shock absorbers are working. You as a student are given some physical properties — how easily the car bounces, how well the shock absorbers soak up bounces — and how the first bounce went — how far the car bounced upward, how quickly it started going upward — and then work out from that what the motion will be ever after. It’s a bit of calculus and you might do it analytically, working out a complicated formula, or you might do it numerically, letting one of many different computer programs do the work and probably draw a picture showing what happens. That’s shown in class, and then for homework you do a couple problems just like that but with different numbers, and for the exam you get another one yet, and one more might turn up on the final exam.
That’s all fine for building one’s ability to do the calculus involved, and maybe the numerical programming too, but it’s also kind of dull. If you can do this problem correctly once you can do it any number of times; we just have students do the problem a couple times over because it takes practice to get it correctly.
But what’s interesting isn’t the exact solution of the exact problem for a particular set of starting conditions. When your car goes over a bump, you’re interested in what the behavior is: is there a sudden bounce and a slide back to normal? Does the car wobble for a short while? Does it wobble for a long while? What’s the behavior?
And this is where phase space gets to be interesting and where you become a graduate student: instead of looking at how high the car is after one shock, look at what the important variables are. In a problem like this, that’ll ordinarily be the position and the momentum of the car. As time goes on, the position and the momentum of the car are going to change, in a way that chains the both of them together along with important physical properties like the energy of the system. You can imagine this — and you should, before your qualifiers — by imagining the position and the momentum as the different axes of a graph, and the system as a point that moves around in a path called either a trajectory or an orbit.
And where this gets really interesting is to imagine a whole bunch of points — positions and momentums for the car — and what happens to that collection as time evolves. Each one of these points represents a different shock that the car can get, and that the shock absorbers try to deal with; but each of them is going to react a little bit differently. The different points will spread out in a way that looks uncannily like bubbles in a rushing stream. Different points might go into different directions, and that represents the original system doing different things — bouncing the one time, for example, or bouncing twice, or bouncing for a long while.
After that, in grad school, you can start looking at a physics problem by trying to identify the different kinds of behavior, such as how many bounces it takes for the car to settle down again, and then working out what combinations of position and momentum put the car in each of those different behaviors, and where the dividing lines between those kinds of behavior are.
ElKement, as might be expected, discusses all this (so if my version doesn’t make sense, perhaps hers will, particularly as she’s got animated pictures to show it), as well as how it starts to change if you start to look into quantum mechanics.