“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.

Complexification

This post may seem a little out there, but that might be the point.

Last week in differential equations we learned about a process our textbook called complexification. (You can go ahead and google that, but near as I can tell what you’ll find is only vaguely related to what my textbook is talking about.) Complexification is a way to take a differential equation that looks like it’s about sines and cosines and instead make it about complex exponentials. What does that mean?

Well, I think most people know a little bit about sine and cosine functions. At the very least, I think most people know what a sine wave looks like.

Shout out to Wikipedia.

Such a wave is produced by a function that looks something like f(x) = sin(x). Sine and cosine come from relationships between triangles and circles, but they can be used to model periodic, fluctuating motion. For example, the way in which alternating current goes back and forth between positive and negative is sinusoidal.

On the other hand, exponential functions don’t seem at all related. Exponential functions look something like f(x) = e^x, and their graphs have shapes such as this:

Thanks again, Wikipedia.

Exponential functions are used to model systems such as population growth or the spread of a disease. These are systems where growth starts out small, but as the quantity being measured grows larger, so too does the rate of growth.

Now, at first blush there doesn’t appear to be a lot of common ground between sine functions and exponential functions. But it turns out there is, if you throw in complex numbers. What’s a complex number? It’s a number that includes i, the imaginary unit, which is defined to be the square root of -1. You may have heard of this before, or you may have only heard that you can’t take the square root of a negative number. Well, you can: you just call it i.

So what’s the connection? The connection is Euler’s formula, which looks like this:

e^ix = cos(x) + isin(x).

Explaining why this formula is true turns out to be very complicated and a bit beyond what I can do. So just trust me on this one. (Or look it up yourself and try to figure it out.) Regardless, by complexifying, you have found a connection between exponentials and sinusoids.

How does that help with differential equations? The answer is that complexifying your differential equation can often make it simpler to solve.

Take the following differential equation:

d²y/dt² + ky = cos(x).

This could be a model of an undamped harmonic oscillator with a sinusoidal forcing function. It’s not really important what that means, except to say you would guess (guessing happens a lot in differential equations) that the solution to this equation involves sinusoidal functions. The problem is, you don’t know if it will involve sine, cosine, or some combination of the two. You can figure it out, but it takes a lot of messy algebra.

A simpler way to do it is by complexifying. You can guess instead that the solution will involve complex exponentials, and you can justify this guess through Euler’s formula. After all, there is a plain old cosine just sitting around in Euler’s formula, implying that the solution to your equation could involve a term such as e^ix.

This idea of complexification got me thinking about the topic of explaining things to people. You see, I think I tend to do a bit of complexifying myself a lot of the time. Now, I don’t mean I throw complex numbers into the mix when I don’t technically have to; rather, I think I complexify by adding more than is necessary to my explanations of things. I do this instead of simplifying.

Why would I do this? After all, simplifying your explanation is going to make it easier for people to understand. Complexifying, by comparison, should make things harder to understand. But complexifying can also show connections that weren’t immediately obvious beforehand. I mean, we just saw that complexifying shows a connection between exponential functions and sinusoidal functions. Another example is Euler’s identity, which can be arrived at by performing some algebra on Euler’s formula. It looks like this:

e^iπ + 1 = 0

This is considered by some to be one of the most astounding equations in all of mathematics. It elegantly connects five of the most important numbers we’ve discovered. Stare at it for awhile and take it in. Can that identity really be true? Can those numbers really be connected like that? Yup.

That, I think, is the benefit of complexifying: letting us see what is not immediately obvious.

It turns out last week was also Carl Sagan’s birthday. This generated some hubbub, with some praising the man and others wishing we would just stop talking about him already. Carl Sagan was admittedly before my time, but he has had an impact on me nonetheless. No, he didn’t inspire me to study science or pick up the telescope or anything like that. But I am rather fond of his pale blue dot speech, to the extent that there’s even a minor plot point about it in one of my half-finished novels.

Now, I read some rather interesting criticism of Sagan and his pale blue dot stuff on a blog I frequent. A commenter was of the opinion that Sagan always made science seem grandiose and inaccessible. That’s an interesting take, but I happen to disagree. Instead, I think we might be able to conclude that Sagan engaged in a bit of complexifying. No, he certainly didn’t make his material more difficult to understand than it had to be; he was a very gifted communicator. What he did do, however, and this is especially apparent with the pale blue dot, is make his material seem very big, very out there. You might say he added more than was necessary.

In doing so, he showed connections that were not immediately obvious. The whole point of his pale blue dot speech is that we are very small fish in a very big pond, and that this connects us to each other. The distances and differences between people are, relatively speaking, absolutely miniscule. From the outer reaches of the solar system, all of humanity is just a pixel.

But there are more connections to be made. Not only are all us connected to each other; we’re also connected to the universe itself. Because, you see, from the outer reaches of the solar system, we’re just a pixel next to other pixels, and those other pixels are planets, stars, and interstellar gases. We’re all stardust, as has been said.

This idea that seeing the world as a tiny speck is transformative has been called by some (or maybe just Frank White) the overview effect. Many astronauts have reported experiencing euphoria and awe as a result of this effect. But going to space is expensive, especially compared to listening to Carl Sagan.

So yeah, maybe Sagan was a bit grandiose in the way he doled out his science. But I don’t think that’s a bad thing. I just think it shows the connection between Sagan and my differential equations class.

For My Next Trick...

I will calculate the distance from the Earth to the Sun using nothing but the Earth’s temperature, the Sun’s temperature, the radius of the Sun, and the number 2. How will I perform such an amazing feat of mathematical manipulation? Magic (physics), of course. And as a magician (physics student), I am forbidden from revealing the secrets of my craft (except on tests and this blog).

During last night’s physics lecture, the professor discussed black-body radiation in the context of quantum mechanics. In physics, a black body is an idealized object that absorbs all electromagnetic radiation that hits it. Furthermore, if a black body exists at a constant temperature, then the radiation it emits is dependent on that temperature alone and no other characteristics.

According to classical physics, at smaller and smaller wavelengths of light, more and more radiation should be emitted from a black body. But it turns out this isn’t the case, and that at smaller wavelengths, the electromagnetic intensity drops off sharply. This discrepancy, called the ultraviolet catastrophe (because UV light is a short wavelength), remained a mystery for some time, until Planck came along and fixed things by introducing his eponymous constant.

Thanks, Wikipedia.

The fix was to say that light is only emitted in discrete, quantized chunks with energy proportional to frequency. Explaining why this works is a little tricky, but the gist is that there are fewer electrons at higher energies, which means fewer photons get released, which means a lower intensity than predicted by classical electromagnetism. Planck didn’t know most of those details, but his correction worked anyway and kind of began the quantum revolution.

But all of that is beside the point. If black bodies are idealized, then you may be wondering how predictions about black bodies came to be so different form the observational data. How do you observe an idealized object? It turns out that the Sun is a near perfect real-world analog of a black body, and by studying its electromagnetic radiation scientists were able to study black-body radiation.

Anywho, my professor drew some diagrams of the Sun up on the board during this discussion and then proposed to us the following question: Can you use the equations for black-body radiation to predict the distance from the Earth to the Sun? As it turns out, the answer is yes.

You see, a consequence of Planck’s law is the Stefan-Boltzmann law, which says that the intensity of light emitted by a black body is proportional to the 4th power of the object’s temperature. That is, if you know the temperature of a black body, you know how energetic it is. How does that help us?

Well, the Sun emits a relatively static amount of light across its surface. A small fraction of that light eventually hits the Earth. What fraction of light hits the Earth is related to the how far away the Earth is from the Sun. The farther away the Sun is, the less light reaches the Earth. This is pretty obvious. It’s why Mercury is so hot and Pluto so cold. (But it’s not why summer is hot or winter cold.) So if we know the temperature of the Sun and the temperature of the Earth, we should be able to figure out how far one is from the other.

To do so, we have to construct a ratio. That is, we have to figure out what fraction of the Sun’s energy reaches the Earth. The Sun emits a sphere of energy that expands radially outward at the speed of light. By the time this sphere reaches the Earth, it’s very big. Now, a circle with the diameter of the Earth intercepts this energy, and the rest passes us by. So the fraction of energy we get is the area of the Earth’s disc divided by the surface area of the Sun’s sphere of radiation at the point that it hits the Earth. Here’s a picture:

I made this!

So our ratio is this: P_e/P_s = A_e/A_s, where P is the power (energy per second) emitted by the body, A_e is the area of the Earth’s disc, and A_s is the surface area of the Sun’s energy when it reaches the Earth. One piece we’re missing from this is the Earth’s power. But we can get that just by approximating the Earth as a blackbody, too. This is less true than it is for the Sun, but it will serve our purposes nonetheless.

Okay, all we need now is the Stefan-Boltzmann law, which is I = σT⁴, where σ is a constant of proportionality that doesn’t actually matter here. What matters is that I, intensity, is power/area, and we’re looking for power. That means intensity times area equals power. So our ratio looks like this:

σT_e⁴4πr_e² / σT_s⁴4πr_s² = πr_e² / 4πd²

This is messy, but if you look closely, you’ll notice that a lot of those terms cancel out. When they do, we’re left with:

T_e⁴ / T_s⁴r_s² = 1 / 4d²

Finally, d is out target variable. Solving for it, we get:

d = r_sT_s² / 2T_e²

Those variables are the radius of the Sun, the temperatures of the Sun and the Earth, and the number 2 (not a variable). Some googling tells me that the Sun’s surface temperature is 5778 K, the Earth’s surface temperature is 288 K, and the Sun’s radius is 696,342 km. If we plug those numbers into the above equation, out spits the answer: 1.40x10¹¹ meters. As some of you may remember, the actual mean distance from the Earth to the Sun is 1.496x10¹¹ meters, giving us an error of just 6.32%.

I’d say that’s pretty damn close. Why an error of 6%? Well, we approximated the Earth as a black body, but it’s actually warmer than it would be if it were a black body. So the average surface temperature we used is too high, thus making our answer too low. (There are other sources of error, too, but that’s probably the biggest one.)

There is one caveat to all this, however, which is that the calculation depends on the radius of the Sun. If you read the link above (which I recommend), you know, however, that we calculate the radius of the Sun based on the distance from the Earth to the Sun. But you can imagine that we know the radius of the Sun (to far less exact measurements) based solely on its observational characteristics. And in that case, we can still make the calculation.

Anywho, there’s your magic trick (physics problem) for the day. Enjoy.

National Hard Things Take Practice Month

Okay, it doesn’t have quite the same ring to it as National Novel Writing Month, but I’m saving my good words for my, well, novel writing. As some of you may know, November is NaNoWriMo, a worldwide event during which a bunch of people get together to (individually) write 50,000 words in 30 days. I’ve done it the last several years and I’m doing it this year, too. It’s hard, it’s fun, and it’s valuable.

As some of you may also know, Laura Miller, a writer for Salon, published a piece decrying NaNoWriMo. (Turns out she published that piece 3 years ago, but it's making the rounds now because NaNo is upon us. Bah, I'm still posting.) This made a lot of wrimos pretty upset, and I’ve seen some rather vitriolic criticism in response. Miller’s main point seems to be that there’s already enough crap out there and we don’t need to saturate the world with more of it. Moreover, she thinks we could all do a little more reading and a little less writing.

Well, as a NaNoWriMo participant and self-important blogger, I think I’m going to respond to Miller’s criticism. Of course, maybe that’s exactly what she wants. By writing this now, I’m not writing my NaNo novel. Dastardly plan, Laura Miller.

Now, I understand the angry response to Miller’s piece. I really do. It has a very “get off my lawn” feel to it that seems to miss the point that, for a lot of people, NaNo is just plain fun. But her two points aren’t terrible points, and I think they’re worth responding to in a civil, constructive way. So here goes.

As is obvious to anyone who’s read this blog, I quite like science. That’s what the blog is about, after all. In fact, I’ve been interested in science ever since I was a child. I read books about science, I had toy science kits, and I loved science fiction as a genre.

Yet this blog about science is not even a year old, and I’m writing this post as a freshly minted 28 year old. Why is that? Because up until about 2 years, I didn’t do anything with my interest in science. I took plenty of science and math classes in high school, but I mostly dithered around in them and didn’t, you guessed it, practice.

It wasn’t until 2 years ago that I sat down and decided it was time to reteach myself calculus. And how did I teach myself calculus? By giving myself homework. By doing that homework. By checking my answers and redoing problems until I got them right. And now I can do calculus. Now I can do linear algebra, differential equations, and physics. I’m no expert in these subjects, but I understand them to a degree because I’ve done them. I’ve practiced, just like you practice a sport.

The analogy here should be clear. You have to practice your sport, you have to practice your math, you have to practice your writing. Where some may disagree with this analogy is the idea that writing 50,000 words worth of drivel counts as practice. The answer is that it’s practicing one skill of writing, but not all writing skills. This follows from the analogy, too. Sometimes you practice free throws; other times you practice taking integrals. Each is a specific skill within a broad field, and each takes practice.

And as any writer knows, sometimes the most difficult part of writing is staring at a blank white page and trying to find some way to put some black on it. We all have ideas. We all have stories and characters in our heads. But exorcising those thoughts onto paper is a skill wholly unto itself, apart from the skills of grammar, narrative, and prose.

So it needs practice, and NaNoWriMo is that practice. If you’re a dedicated writer, however, then it follows that NaNo should not be your only practice. You have to practice the other skills, too. You have to write during the rest of the year, and you have to pay attention to grammar, narrative, and prose. But taking one month to practice one skill hardly seems a waste.

I’ve less to say about Miller’s second point, that we should read more and write less. This is a matter of opinion, I suppose. But I do have one comment about it. America is often criticized as being a nation of consumers who voraciously eat up every product put before them. We are asked only to choose between different brand names and to give no more thought to our decisions than which product to purchase.

Writing is a break from that. Rather than being a lazy, passive consumer of other people’s ideas, writing forces you to formulate and express your own ideas. Writing can be a tool of discovery, a way to expand the thinking space we all inhabit. Rather than selecting an imperfect match from a limited set of options, writing lets you make a choice that is precisely what you want it to be. You get to declare where you stand, or that you’re not taking a stand at all. You get to have a voice beyond simply punching a hole in a ballot.

You shouldn’t write instead of read, but you should write (or find some other way to creatively express your identity).