Five years ago today I posted, Beautiful Math, which is about Euler’s Identity. In the first part of that post I explored why the Identity is so exquisitely beautiful (to mathematicians, anyway). In the second part, I showed that the Identity is a special case of Euler’s Formula, which relates trigonometry to the complex plane.
Since then I’ve learned how naïve that post was! It wasn’t wrong, but the relationship expressed in Euler’s Formula is fundamental and ubiquitous in science and engineering. It’s particularly important in quantum physics with regard to the infamous Schrödinger equation, but it shows up in many wave-based contexts.
It all hinges on the complex unit circle and the exp(i×π×a) function.
As you’ll see, this post brings together many previous posts:
- Beautiful Math, which introduces Euler’s Identity and Formula. (For this post, it’s important that you’ve read this one.)
- Many posts about complex numbers, and in particular the complex plane. (In this post you do need to know what the complex plane is.)
- Fourier Curves, which introduces some Fourier transform ideas, including that it’s used in JPEG compression. (Optional reading.)
An irony about that last post is that, except for the title and image captions, I don’t actually mention Fourier transforms at all. (It was mainly about the pretty pictures.) In the next two posts I plan to show you how Fourier transforms work.
That — and more — is based on the following:
One might see just an opaque bit of math. One might recognize the mathematical operation of exponentiation — something to the power of something. One might even recognize the constants e and i but not have any sense of their appearance, use, or implications, here.
The point is that the expression might look mysterious, but it doesn’t look surprising.
On the other hand, if one is more familiar with the constants and the operation, one might very well find the expression not just mysterious, but surprising. And weird! There may be a sense of WTF? or that it can’t work (or at least how does that work?).
The level of hey wait a minute may only increase with this:
What in the world can it possibly mean to raise the transcendental number, e, to the power of i times π? How is that even possible? And how can that be equal to minus one?
One clue is that we’re using the imaginary unit, i, which means the complex numbers, which means the complex plane, which means there is a geometric aspect to this (and it’s the geometry that’s cool and makes Fourier transforms work).
To make sense of it, we must extend the definition of exponentiation from the basic one most of us first learned.
That definition saw the exponent as an integer specifying how many times to multiply the base number times itself. For instance:
We multiply the base number, N, times itself five times. Simple, but as we got a little deeper into math, we did encounter other definitions:
Note that all three of the above equalities can be mathematically justified using the hyper-important exponent rule:
There are also logarithms, which use real exponents, which gets us closer to the idea of something like π as an exponent.
(Regarding pi, 10π is just the number 1,385.455731… — as with the logarithm above, pi is just another real exponent with a value a bit more than three. It’s using i as an exponent that’s the weird addition here.)
The point here is that, even on the outskirts of math, exponents are part of a larger picture, some of which you’ve probably already seen.
With that introduction, here’s a very good (and very short) video that introduces the ideas behind what at first seems an impossible notion:
(Note that 3.14 minutes is not the same as 3:14 minutes!)
The Executive Summary: In the extended definition, exponentiation with i becomes rotation on the complex plane. When we choose to use e as the base, a full rotation of 360° is exactly 2π radians, which is the circumference of a circle and which therefore allows a direct mapping to trigonometry.
(As I’ve posted about not long ago, multiplication by i rotates points by 90° on the complex plane. See also: Matrix Rotation)
This mapping to trig is where Euler’s Formula comes in:
eia = cos(a) + i sin(a)
Both sides of the expression are ways to denote a complex number that lies on the complex plane at angle a. As you might imagine, treating a complex number as a single exponential value offers some calculation advantages.
(Euler’s Identity, eiπ = -1, is the specific case where a=π.)
But why e? What’s so special about that number? This (14-minute) video explains:
The short version is that the function ex is its own derivative, which tames certain aspects of the math.
As the video gets into, derivatives are important throughout physics (and other areas of life, such infection rates and your bank account). We encounter derivatives everywhere; velocity, for example, is the derivative of distance over time (as seen in the first video).
In general, the function ax (for some value of a) is proportional to its derivative, but only when a=e is the derivative equal. (Remember that e is just a transcendental constant with the value 2.7182818284590…)
The upshot is that, when e is the base, there are useful manipulations of formulas containing it (as seen in both videos). That it’s the base of the natural logarithm makes it a natural choice in many situations (as, for instance, the unit circle on the complex plane).
As far as how the calculation is actually carried out, the short form is:
The exp(x) function is the same thing as raising e to the power of x. It makes the expression easier to write and typeset. It also makes it more clear that “e-to-the-x” is a function.
That short form expands to:
For however large we make n. (The larger it is, the more accurate the calculation.) Note that x can easily be a complex number in this expression.
Just remember: ex = exp(x)
Let’s start with the general exponential function.
Figure 1 shows a set of exponential curves.
Each curve plots ax for some value of a.
The blue curves have values for a above one, and the purple curves have values below one.
The red curve (flat at one) is a=1.0. (One to the power of anything is just one.)
The green curve is a=e.
The dark-blue curve is a=1.5. I included it to show how the flat a=1.0 curve starts to deflect upwards for positive values of x (and downwards for negative values).
The three blue curves, from outside-in, are a=2.0, a=4.0 & a=8.0. (Notice the green a=e curve is between the 2.0 and 4.0 curves — the value of e is 2.71828…)
The purple curves, from outside-in, are a=0.50, a=0.25 & a=0.125. When values for a increase above 1.0, the curve deflects more and more sharply upwards (after passing through 1.0 at x=0.0). For values of a less than 1.0, the curve deflects more sharply as the value decreases towards zero.
All curves pass through 1.0 at x=0.0, because a0 is always 1.0.
There is a bit of magic we can apply regarding the base value, a:
In other words, ax is the same as exk for some constant k — specifically the natural log of a. So for example, if a=3:
This means we can draw an identical set of curves as Figure 1 using just the exp(x) function (which is exactly how I did it).
We can also leverage that nice derivative property and unify our mathematical approach. So it’s the right-hand version that usually appears in physics formulas. That’s part of why it shows up so often.
The exp(x) function does much more than create “exponentially rising” curves. Here is another set of curves drawn with the exp(x) function:
You may recognize these as “bell curves” — their formal name is Gaussian (after mathematician Carl Gauss). They are described by the Gaussian function:
Which, remember, is the same thing as:
The constants A, B & C, control the shape of the curve. The value of A determines the height of the peak; the value of B controls where it is centered, and the value of C (which must be greater than zero) controls its width.
In Figure 2, the value of A is 1.0, the value of B is 0.0, and the various curves reflect different values of C.
(Gaussian curves show up a lot in quantum mechanics as the localization of momentum, energy, or position, to name a few. This is closely tied to what comes next.)
Now consider the following double-chart, which shows two ways of representing the same thing:
Both show a combination (or superposition) of three sine waves at three different frequencies: 13, 27, and 42 cycles per second. The upper chart shows the energy at given times, and the lower chart shows energy at given frequencies.
The upper chart averages three copies of the exponential function, each generating a different frequency sine wave (note the use of i here):
The lower chart averages three copies of the Gaussian function, each centered on one of the frequencies:
Two ways of looking at the same thing. The lower chart represents a Fourier Transform of the waveform. The upper chart can be constructed from frequency information via a Inverse Fourier Transform.
The Fourier transform is a fundamental tool in wave physics.
As one example, audio spectrum analyzers are common among audio engineers and enthusiasts, because they display, in real time, how much energy (sound) exists at different frequencies. (They’re also fun to watch.)
After good old VU meters, they are one of the more common audio displays (see an example image to the right).
Such analyzers use a Fourier transform to break an incoming signal — which is energy varying in time (the upper chart in Figure 3) — into its component frequencies. (The image here is essentially the same thing as the lower chart in Figure 3.)
In quantum mechanics, Fourier transforms are involved in solutions to the Schrödinger equation, and they underpin the Heisenberg Uncertainty Principle.
That’s where I’ll pick up next time.
For this and many other posts I’m deeply indebted to Grant Sanderson and his YouTube channel, 3Blue1Brown. If you have any interest in math, you should to subscribe to this channel. It’s by far the best math illumination channel I’ve encountered.
I owe a lot of my recent progress to this guy and his videos!
Stay exponential, my friends. Go forth and spread light and beauty.
October 17th, 2020 at 9:25 am
I mentioned justifying the equalities using the exponent rule:
There is also a rule: N1 = N. (We saw that in how all the simple exponential curves passed through 1.0.)
So for this equality:
N5 = N1+1+1+1+1
N5 = N1 × N1 × N1 × N1 × N1
N5 = N × N × N × N × N
The next two are ever so slightly more involved:
For the first one, first consider that, trivially:
N0+x = Nx
N0 × Nx = Nx
Therefore (dividing both sides by Nx):
N0 = 1
Justifying that basic identity. With that in mind:
Nx-x = N0 = 1
Nx × N-x = 1
Dividing both sides by Nx:
N-x = 1/Nx
With those examples, for now I’ll leave the last one for the interested reader. This second one was a bit involved — the third one is a lot easier.
October 17th, 2020 at 9:31 am
If you want to get a bit deeper into Euler’s formula, here’s a video Grant did during the Lockdown phase in the first half of this year. It’s a bit long (51 minutes) and oriented mostly towards a high school math level, but it’s worth watching if you have an interest.
October 17th, 2020 at 9:33 am
One more. This one is definitely “optional reading” but it again explores Euler’s ubiquitous formula and also introduces group theory, which is another major part of mathematics (it’s 24 minutes long):
October 17th, 2020 at 6:01 pm
As usual, nothing mathy to say, except I scanned this one a little more carefully since you mentioned the Schrodinger equation. I get more interested in math when it’s about something.
October 17th, 2020 at 6:44 pm
Heh! Part of me wants to protest that math is always about something, but I know what you mean. These two posts (one coming tomorrow) have a direct application to quantum mechanics. If you look at the Wiki page for the Schrödinger equation you don’t have to go very far down the page to find the first formula with e-to-the-i, and it reappears constantly throughout the page.
It really is a fundamental building block, and having a grasp of what it means opens a lot of doors in physics. (Not just QM, but optics and sound.) It also open a door to the Fourier transform, which turns out to be another basic building block.
FWIW, I have a major commitment to the idea of foundation knowledge. On a practical level, it’s what allowed me to change modes during a changing career. Good foundations make it easier to build new knowledge. An analogy I like is that, if one knows music and has already learned to play an instrument, learning a new instrument is just a matter of learning how to operate the new thing. The more instruments one plays, the easier new ones become.
Math is one of those foundations that tends to show up in many of my favorite places, so I’ve always wanted to know more, and — fortunately for me — I find math fascinating, so it’s easy and rewarding to pursue.
The downside of foundation knowledge — one reason I think a lot of people lack it — is that the ROI really sucks plus it takes forever to acquire. It’s exactly the reaction you’re expressing to math — what’s this for and what’s the point? It’s not very satisfying to say, “Well, maybe it will come in handy later,…” It just takes being fascinated by something I guess. (Or discipline I lack, since otherwise I’d know a lot more about chemistry and biology.)
I think there are a number of things with shallow learning curves — long stretches of effort with little payoff. Music was that way for me long, long ago. Then one reaches an inflection point in the curve and progress skyrockets. Suddenly the pieces makes sense and fit together.
At least that’s how a lot of things have worked for me. Long shallow curve with a steep slope after the light bulb finally goes on. (I’ve gotten the impression my curve is shallower than many in the beginning, but I make up for it later.)
October 17th, 2020 at 7:18 pm
I’m usually a foundations guy myself. In school, I was always at an disadvantage early on, because while others were just memorizing heuristics, I was trying to grok the concept at a deeper level. Of course, doing so always eventually translates into advantages later on.
I’m gradually listening my way through Lex Fridman’s interview of Scott Aarsonson (Aaronson links to the Youtube version from a recent blog post). One of the things he discusses is how understanding a few key concepts enables a lot in computer science. I found his take interesting.
The problem is that, with math, I’ve just never felt the bug, never been interested in it for its own sake. And when I attempt to dig too much into it, it awakens anxieties from all the struggles I had in school. Math books always seem to be written by someone who thinks it, in and of itself, is inherently beautiful. Well, I’m sorry to say it isn’t beautiful to me. I need to have it relentlessly mapped back to something useful, or it takes enormous discipline for me to continue.
I’ve struggled even getting through a chapter on linear algebra in a quantum computing book, despite knowing that I need it for the rest of the book, because the author switches into pure math mode, shoveling it without mentioning how any of it maps to the quantum subject. (The reading I did do wasn’t totally in vain. I noticed that some of the notation in quantum physics papers suddenly seemed less cryptic.)
October 17th, 2020 at 8:26 pm
Sounds like we have similar approaches to learning something. I need to understand — I can’t learn by rules or facts. As you say, it pays off big time later.
I think a lot of people got disenchanted with math in school. It can be a very poorly taught subject, sometimes by teachers who don’t grasp or love it themselves. As you’ve found from books, just loving it isn’t enough — that doesn’t carry over to others. It takes connecting it or illuminating it in a compelling and engaging way. (That’s why I love that 3Blue1Brown channel — that’s what the guy does, and he’s really good at it.)
If you wanted to tackle linear algebra again, I highly recommend his Essence of linear algebra series. It’s 15 videos ranging from the 4-17 minute range; most of them kind of in the middle of that. A lot of working with the Schrödinger equation involves concepts from linear algebra, eigenvectors and eigenvalues key among them. The series involves a visual understanding of matrix transformations that makes them almost obvious.
(Tomorrow’s post involves a video of his that makes the Fourier transform almost obvious. It’s a really cool way to look at it, and this e-to-the-power-of-i business is literally at the heart of it. And as I may have mentioned, the Fourier transform underpins the HUP — position and momentum are conjugate Fourier transforms of each other.)
Exactly true about the math looking less cryptic! More and more that quantum math is starting to look like something I understand, although I’m a long ways from being able to work with it.
October 18th, 2020 at 4:36 pm
I’m not kidding about synchronicity permeating my life. Today I wanted to curl up with a book, so I borrowed an immediately available Rex Stout book from the library. I didn’t check my collection of purchases closely enough — I already own the book.
So while reading it seems more and more familiar, and I’m trying to figure out if some TV show borrowed the plot or why this seems so familiar. I finally checked my collection more closely and, sure enough, it’s familiar because own it and I’ve read it. (Within the last couple of years, at that. As I’ve said many times, my memory for fiction plots is practically non-existent.)
But here’s the eerie part. One character is a mathematician and in one scene he writes out a formula for what he calls the “second approximation to the normal distribution”. That formula is in the book, and our friend e-to-the-power-of-something is part of that formula. Specifically, it was a Gaussian. (When I say it’s ubiquitous and appears everywhere, I’m not kidding about that, either.)
See this Wiki article for examples.
December 31st, 2020 at 7:59 am
[…] It turns out YouTube is a wonderful resource for physics lectures. There are various QM courses and lectures. A lot has been over my head at first, but by chewing and chewing on it, it starts to come clear. […]
February 8th, 2021 at 7:48 am
[…] (and in wave mechanics in general). [Especially the exponential form (right-hand side above). See Circular Math and Fourier Geometry for […]
August 6th, 2021 at 6:33 am
[…] can also use the exponential form [see: Circular Math and The exp […]
April 7th, 2022 at 7:07 am
[…] note the i, the imaginary unit. Here again, complex numbers are important in quantum math. [See Circular Math and its many links (especially Beautiful Math) for […]