Friday, November 22, 2013

“We won’t go into the details.”

I’m pretty sure I’ve talked a lot more about my math class than my physics class this semester. With the semester winding to a close, I don’t have much time life to even the score. But here’s an attempt. The reason for the relative silence on the subject of physics is, however, math-related.

As I mentioned in an earlier post, the third semester of intro physics is usually referred to as modern physics. At my community college, it’s “Waves, Optics, and Modern Physics.” The course covers a lot of disparate material. While the first half of the semester was pretty much all optics, the second half has been the modern physics component.

What does “modern physics” mean? Well, looking at the syllabus, it means a 7-week span in which we talked about relativity, quantum mechanics, atomic physics, and nuclear physics. All of these are entire fields unto themselves, but we spent no more than a week or two on each topic.

I predicted during the summer that I wouldn’t mind the abbreviated nature of the course, but that prediction turned out to be wrong. Here’s why.

The first two semesters of physics at my community college were, while not perfect by any stretch of the imagination, revelatory in comparison to the third semester. I enjoyed them a great deal because physical insight arose from mathematical foundations. With calculus, much of introductory physics becomes clear.

You can sit down and derive the equations of kinematics that govern how objects move in space. You can write integrals that tell you how charges behave next to particular surfaces. Rather than being told to plug and chug through a series of equations, you’re asked to use your knowledge of calculus to come up with ways to solve problems.

This is in stark contrast to what I remember of high school physics. There, we were given formulas plucked from textbooks and told to use them in a variety of word problems. Kinetic energy was 1/2mv2, because science. There was no physical insight to be gained, because there was no deeper understanding of the math behind the physics.

And so it is in modern physics as well. The mantra of my physics textbook has become, “We won’t go into the details.” Where before the textbook might say, “We leave the details as an exercise for the reader,” now there is no expectation that we could possibly comprehend the details. The math is “fairly complex,” we are told, but here are some formulas we can use in carefully circumscribed problems.

It happened during the optics unit, too. Light, when acting as a wave, reflects and refracts and diffracts. Why? Well, if you use a principle with no physical basis, you can derive some of the behaviors that light exhibits. But why would you use such a principle? Because you can derive some of the behaviors that light exhibits, of course.

But it’s much worse in modern physics. The foundation of quantum mechanics is the Schrödinger equation, which is a partial differential equation that treats particles as waves. Solutions to this equation are functions called Ψ (psi). What is Ψ? Well, it’s a function that, with some inputs, produces a complex number. Complex numbers have no physical meaning, however. For example, what would it mean to be the square root of negative one meters away from someone? Exactly.

So to get something useful out of Ψ, you have to square it. Doing so gives you the probability of finding a particle in some particular place or state. Why? Because you can’t be the square root of negative one meters away from someone, that’s why. The textbook draws a parallel between Ψ and the photon picture of diffraction, in which the square of something also represents a probability, but gives us no mathematical reason to believe this. Our professor didn’t even try and was in fact quite flippant about the hand-waving nature of the whole operation.

If you stick a particle (like an electron) inside of a box (like an atom), quantum mechanics and the Schrödinger equation tell you that the electron can only exist at specific energy levels. How do we find those energy levels? (This is the essence of atomic physics and chemistry, by the way.) Well, it involves “solving a transcendental equation by numerical approximation.” Great, let’s get started! “We won’t go into the details,” the textbook continues. Oh, I see.

Later, the textbook talks about quantum tunneling, the strange phenomenon by which particles on one side of a barrier can suddenly appear on the other side. How does this work? Well, it turns out the math is “fairly involved.” Oh, I see.

This kind of treatment goes on for much of the text.

Modern physics treats us as if we are high school students again. Explanations are either entirely absent or sketchy at best. Math is handed down on high in the form of equations to be used when needed. Insight is nowhere to be found.

Unfortunately, there might not be a great solution to this frustrating conundrum. While the basics of kinematics and electromagnetism can be understood with a couple semesters of calculus, modern physics seems to require a stronger mathematical foundation. But you can’t very well tell students to get back to the physics after a couple more years of math. That’s a surefire way to lose your students’ interest.

So we’re left with a primer course, where our appetites are whetted to the extent that our rudimentary tools allow. My interest in physics has not been stimulated, however. I’m no less interested than I was before, but what’s really on my mind is the math. More than the physics, I want to know the math behind it. No, I’m not saying I want to be a mathematician now. I’m just saying that I can’t be a physicist without being a little bit a mathematician.

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