In my physics class we’ve just recently introduced perturbation theory as a way of solving transcendental equations approximately. The problem is, I don’t have a very good grasp on how it works, or when/how to use it. I have a worked out example (from my notes) that I don’t quite understand, and I’m not certain I’ll be able to speak to the professor before the test on Thursday, and google isn’t so helpful, so if anyone could give me an explanation or possibly a simple worked-out example that would be really helpful.
I don’t know exactly what area of physics you’re studying, so I’ll try to be very general here.
Basically, the idea is like power series in Calc 2 (you have had Calc 2, haven’t you?) That is, given a sequence a[sub]n[/sub] of real numbers, there’s a power series
converges to an actual real number. Then, in the interval (-R,R) we have an honest function, which analytic techniques can tell us a lot about.
Perturbation theory starts with a good approximation to some quantity, neglecting some correction term which is about size d. Then it adds the corrections as a multiple of d, then it adds corrections to those, which are now about d[sup]2[/sup], and so on. See how we’re getting something like a power-series expansion?
Anyhow, if we assume that d is “small enough”, then this series will converge. Further, we can get estimates on how much error we introduce by stopping after the nth step (one of those analytic techniques from above).
One common use is in quantum electrodynamics, where the probability amplitude for some collection of electrons and photons coming in being seen as another collection going out of an experiment is calculated something like this.[1]
Electrons and photons interact two electrons and one photon at a time. So, draw a graph (the “collection of lines and vertices” kind) with each line being labelled as an electron or a proton so that each vertex has two electron lines and one photon line coming in, and with the specified number of electron and photon lines running off the bottom and top of the page. Of course, there are many ways you can do this. Each one models a way that the particles could interact between the bottom and top of the page. To find the probability amplitude overall, we have to find the amplitude for each graph and add them all up.
Calculating these amplitudes involves a lot of messy integrations, but the important thing is this: to each vertex there’s attatched a factor called the “coupling constant” for electrodynamics, which is about 1/137. That is, you can do all these messy integrals for each graph with n vertices and add them up. That’s your nth-order correction term, and the d (measuring how big each correction is compared to the previous) is 1/137, which is pretty small.
[1] Physicists, I know I’m being fast and loose with the details. How do you like the tables being turned on you?
It would be very helpful in answering this question if you were to state what kind of physics class this is. If this is your first encounter with perturbation theory, I’m guessing it’s not QED – maybe undergraduate quantum mechanics? If so, you’ve already probably spent a lot of time solving Schrodinger’s equation for certain potentials, such as the harmonic oscillator or the square well. The simple answer to when to use perturbation theory is when you have a problem where your potential is close to a problem you can solve exactly, but with a little extra part. Say, a square well with a little bump in it. What do I mean by a little extra part? Well, I mean suppose your potential is V + V’, where you could solve the problem exactly if the potential were V alone. If the dominant contribution to the energy spectrum is what you get from V alone, then it’s reasonable to treat V’ as a perturbation. In particular, when solving time dependent problems in quantum mechanics it is often useful to treat the time dependent part of your potential as a perturbation (if it’s a small contribution compared to the static part, that is.)
Anyway, I’m not going to type a whole example problem, but I’d recommend David J. Griffiths Introduction to Quantum Mechanics as an undergrad-level QM textbook with a good explanation of perturbation.
Regardless of whether you’re doing quantum mechanics, a good rule of thumb for whether you can use perturbation theory is if you have a problem where it can be broken down into a problem you can solve exactly plus an additional contribution, and the exactly solveable part is the dominant contribution. In QM, this means a potential that looks like one you can solve exactly, with a little addition. For QED, this means a problem (say, finding the cross section of some process), where the dominant contribution comes from the exactly solvable tree level diagram, and higher order diagrams contribute in successively smaller amounts.
The same theory holds. The idea is that you have a very complex nonlinear function that you need to repeatedly evaluate in a local region. Thus, you assume a steady state with linear variations off of it.
This is very helpful in such fields as aerospace engineering where you may have an aircraft in steady level flight. This is the steady state. You then linearize the stabiliy equations about this steady state. So:
The terms a and b define the stability of this system. This method is most useful when you are numerically solving a series of equations with no explicit solution.
I won’t do you the injustice of offering my bumbling explanation of perturbation theory, but I found the appendix in this book to have a particularly useful brief introduction to perturbation theory. (It’s in an appendix in the back.) I haven’t used this one or this one so I can’t speak to them, but in general I’ve found the inexpensive Dover texts (which are typically reprints of “classic” textbooks in the field) have generally been of much higher content quality and clarification than the $100+ error filled hardcover textbooks I purchased for classes.
I’m guessing you’re taking either statistical mechanics or a second semester QM course. Either way, good luck onto ya. Remember, it’s not how much you know…it’s how little everybody else knows in comparison that counts.
Thanks for the help. I got to speak to my professor after all today so I’ve gotten a better idea of how to do perturbation in the context it arises in my class (intermediate mechanics, by the way.) This thread was helpful so thanks everyone, and I will check out those books Stranger.
Thanks, and don’t be surprised if this thread gets revived next semester when I take Quantum :eek:
