My family and I play a lot of Scrabble, which is a word game where you have to use tiles with letters on them to make words on the board. The board containes multipliers that also give you more points, as well as the tiles themselves, which each have a certain number of points associated with them.
In Scrabble, you usually want to start first, because the first person to play gets to double the points they make. For example, if they played the word FISH, which gives a total of $4+1+1+4=10$ points, their score would be twenty points. As such, being first is a nice boost to get you started in the game.
Recently though, one of my family members had two games where they were the first to start, and during both of these games, they couldn’t actually play a word! This is because there were no vowels in the initial seven tiles they picked. Consequently, they couldn’t play a word, since we don’t recognize any word that doesn’t contain a vowel. This meant that they had to skip their turn on both occasions, which obviously frustrated them.
This got me thinking: how likely was this to happen? In other words, was this event something that happened oftem? I couldn’t remember having this happen before this brief spell of bad luck, so I thought this would be a good application of conditional probability.
There are 100 tiles in total in the game of Scrabble. Out of those 100 tiles, 54 of them are consonants (I’m counting the two blank tiles as vowels, since they can be used as vowels if needed). Then, if we let $A$ be the event that I don’t pick any vowels in seven letters, then this is equal to saying that I want to pick only consonants. If we apply some basic counting principles from discrete mathematics, then we know that the total number of ways to get seven consonants from fifty-four tiles is $54 \choose 7$. Similarly, the number of ways to pick seven letters from the total bag of tiles is $100 \choose 7$. Finally, since it is probably safe to assume that there’s an equally likely chance of picking any given tile, the probability of picking only consonants is then: In other words, there’s about a one-in-one-hundred chance of being unlucky and not being able to start the game. I’d say that this probably matches fairly well with my experience, though I do seem to have noticed this happen more as of recently. Perhaps I’m simply seeing events now that I’ve focused on them.
What is a Differential Equation?
For those who don’t study mathematics but have an interest in science, the term “differential equation” will often come up in conversation. They are often said to be really important to all scientific disciplines, but what are they exactly, and how do they help us describe the world?
To begin with, a differential equation can be thought of as an equation that links a quantity with the rate of change of that quantity (and potentially, the rate of change of the rate of change, and so on). More technically, if we have a function that depends on $x$, we call it $f(x)$. Then, a differential equation could be of the form: Students in physics will perhaps recognize this differential equation as the equation describing simple harmonic motion. For everyone else, the above equation simply relates the function $f$ with its second derivative (which is a measure of how the change in $f$ changes). And since the second derivative describes how the function $f$ changes in some way, this function can be used to describe certain motion (such as that of a spring or a simple pendulum). Of course, if this is the first time you have seen this differential equation, this won’t be obvious to you. However, this is indeed related to those two physical systems, and can actually be seen from another differential equation, namely: This is simply Newton’s second law in the $x$ direction. From this differential equation, one can derive all sorts of equations of motion for classical systems, such as the equations of motion for the spring or the simple pendulum, shown above. This is very powerful, because we are able to work out solutions to most classical mechanics problems using just the above equation (technically, by using one for each axis of motion). Of course, I’m not saying that we can always solve these equations exactly. In fact, most of the time one needs to use a computer to solve the equations numerically, since no analytical form is possible (which means we won’t get a nice formula at the end for the motion).
This brings us to the question of solving these differential equations. What exactly are we trying to solve for in these equations? We are are looking for the specific function $f(x)$ such that we can substitute this function back into the equation and have the equation be satisfied. This turns out to be more difficult than solving regular equations in algebra class, because now we have to deal with the fact that a function and its rate of change (and other derivatives) have to all be satisfied in the differential equation. As such, there are whole courses devoted to studying techniques to finding solutions to classes of differential equations.
However, it gets even more tricky.
I recently wrote about toy models, which are the models we begin with when studying phenomena in science. They are usually simple, and don’t describe reality well. Then, as one increases the complexity of the model, we get a better description of reality. However, in the process, the equations become more difficult to solve. What we have been looking at so far are examples of ordinary linear differential equations. The “ordinary” part means that the functions only have one variable in them (for our first example, the variable was $x$). The “linear” part means that the functions aren’t multiplied by other functions of $x$. The following would not be considered a linear differential equation: The above equation isn’t linear because it has functions multiplied together within the various terms. This doesn’t meant the differential equations are impossible to solve. However, if they do have a solution, they tend to be much more tricky to find.
To give a small example of how quickly we can ramp up in complexity, consider the Schrödinger equation in quantum mechanics for a single particle of mass $m$ in one dimension: This is now a much more involved differential equation. First of all, note that the function that we are trying to solve for, $\Psi(x,t)$, is a function of two variables. This automatically turns this into a partial differential equation, since we have a function of multiple variables. Additionally, if we look at the function $V(x,t)$, then if it isn’t a constant function, it will multiply $\Psi(x,t)$, making the differential equation non-linear as well. This is particularly bad if you are searching for analytic solutions. This is why only some systems in quantum mechanics can be solved exactly (or very close to exact), while most need to be solved numerically.
The equation itself though is how we study quantum mechanics. It’s the backbone of our theory, and describes the time evolution of a system. As such, this partial differential equation is absolutely necessary, even if it can be tricky to solve.
There are many, many more examples of differential equations in the sciences. From the flow of heat to the growth of populations, differential equations describe the world around us. The key reason that scientists use differential equations then, is because differential equations capture motion in between variables. We can get situations where changing one variable bit does not only change the other variable, but also affects how fast that variable will change, and so on. In essence, we get relationship that are more intertwined, and better model the world around us.
What Are Toy Models in Physics?
If you’ve never taken a quantum mechanics class, you might automatically think that one begins studying all of the quantum strangeness that occurs during experiments. Perhaps you jump into complex calculations involving quantum field theory, or perhaps you explore the phenomenon of quantum entanglement.
Similarly, if you’ve never taken a course in classical mechanics, you might think that one studies systems such as a bicycle rolling on the road, and how the tires interact with the ground in a complex way.
However, this is far from the truth.
In quantum mechanics, one begins by studying probability and the Schrödinger equation in different cases. Quantum field theory isn’t even in the cards for the first few courses. For classical mechanics, complex situations such as a bicycle on the road aren’t approached until the tail end of a course. If asked, a professor will probably say something along the lines of, “We aren’t studying this now, because it involves more complicated terms such as friction.”
Instead, here are the types of systems that are studied. For quantum mechanics, systems such as the free particle, the particle in a box, the quantum harmonic oscillator, and the finite square well are studied. For classical mechanics, blocks sliding on planes and other frictionless surfaces, pendulums (but only for small angles), and spring systems are typically studied.
Hearing this, a question might naturally arise. If we aren’t even studying realistic systems, how can we learn anything about the reality around us?
A lot, it turns out.
The examples I gave above are what physicists like to call “toy models”. They are named as such because they exist to exhibit certain properties of a system without complicating matters unnecessarily. Much like mathematics does when it generalizes and abstracts concepts in order to be more encompassing, physicists study and learn about toy models because they are useful mental models to have when moving to more complicated systems.
To give you an idea of how these systems can actually be incredibly useful later on, let’s look at what seems like an innocent example from quantum mechanics: the particle in a box.
First, a small preamble on quantum mechanics. We study the quantum world through the Schödinger equation, which is given by the following differential equation: Here, $E$ is the energy of the particle that we are looking at, and $V$ is its potential energy. Using this equation, the goal is always to solve for $\psi$ and $E$. These are two important quantities in quantum mechanics. The energy is something we can easily imagine, but the wavefunction $\psi$ might not be as clear to readers.
On its own, $\psi$ isn’t anything special in terms of physics. It’s only once we take the magnitude of the wavefunction, denoted $\vert \psi \vert ^2$, that we get a quantity that can be interpreted physically. The interpretation is that is it the probability density of the particle, in either its coordinate-representation (its location in space), or in its momentum-representation (the momentum the particle can possibly have). Roughly speaking, the probability density gets integrated over a small volume in order to give a probability that the particle will either be in a small region of space or have a particular range of momentum (the equivalent to a region of space). Therefore, the probability density isn’t exactly a probability, but it is definitely related.
Back to our example. The particle in a box is one of the simplest systems one can study. It involves letting a particle live in a box and determining how the wavefunction $\psi$ evolves over time. How do we do that? Well, we set the potential $V$ to be infinite when one is outside of the box, and finite (in the usual case, zero), when one is inside the box. An infinite potential is akin to saying that $\psi = 0$ outside of the box.
Inside the box, the potential is then $V = 0$ (usually), which means we can solve the above equation to figure out both the energy $E$ and the wavefunction $\psi$. The actual derivation of these two quantities isn’t terribly important here, but one consequence in particular is worth mentioning: the energy of a particle comes in discrete chunks. In other words, a particle which is confined to a box can’t have any old energy that it wants. There are specific values allowed, and no others.
Obviously, this model isn’t a realistic description of our reality. However, it does have some useful features. For one, it illustrates very simply some of the strange properties of quantum mechanics. As soon as a particle is confined in some way (in this case, by a box), the particle goes from being able to have any energy to having only specific possible values for its energy. This is true for the particle in a box, but it is also true for other systems. Think about that. As soon as you introduce some kind of confining potential to a particle, its energy goes from being a continuous quantity to being discrete. This tells us something very important about quantum mechanical systems! (Of course, this would need to be proved in more generality, but it gives us a taste of one consequence in quantum mechanics.) It just so happens that this particular system is very simple, so it’s one that is easily studied to learn about some properties of quantum mechanical systems.
The same is true for the study of classical systems, such as the pendulum. Here, we make a bunch of approximations that render the system unrealistic, but “close enough” to the real thing that it is still useful to study. These approximations include the string or rod holding the pendulum being massless, the bob at the end of the pendulum having all of its mass concentrated to a point, and the amplitude of the pendulum being only a few degrees. If you relax these approximations, then you get a more realistic system, but the equations are much more difficult to handle.
Therefore, we make do with our toy models. We know that they don’t tell the whole story, but they give us a good idea on how a system will behave. Additionally, these toy models can be used to initially study a complex system, and slowly add new “features” to the toy model in order for it to better mirror the system we want to study. You can think of it as upgrading a computer system with better components. As such, the new computer will better handle the tasks you want it to do, just like the new toy model will more accurately reflect the system we want to study.
Despite the name, toy models aren’t useless mathematical equations that physicists write down as a joke. Their purpose is to instruct, allowing one to study a complex system in its broadest strokes. We take a complex system, try and distill it down to its essential parts, and then come up with a toy model of it. This toy models simplifies things, but the point is that we can eventually reintroduce the complexity back in, slowly building up to our original system once again. It may have been impossible to initially study our system of interest mathematically, but by breaking it down and then building it back up, we learn a lot about the essence of a system, and therefore, we can gain insights that weren’t apparent to us initially.
A Simple Puzzle
I love simple puzzles with elegant solutions that are perhaps unexpected. I particularly enjoy it when these puzzles can be solved without invoking a whole bunch of machinery to support the solution. Mathematical machinery is nice, but a great simple puzzle can be a great segway into that machinery. As such, invoking it right off the bat kind of misses the point.
For this post, here’s the puzzle I’m thinking about:
Can you find two squares of lengths $a$ and $b$ which have an area equal to a square of length $(a+b)$?
If you prefer a diagram version of this problem, here it is.
At first, this puzzle seems simple enough. How hard can it be to match these conditions? But after a few attempts, you may start to get frustrated, because they don’t seem to work. It’s as if the question is deliberately trying to work against you, thwarting each attempt. The right combination just doesn’t seem to be there. What’s up?
After a while, you may start to think that the answer to this puzzle, is no, it cannot be done. To show this, you might start to wonder what the area of the two squares is, and what the area of the larger square is. Then, by comparing them, we can see if there’s some way to make them equal (or to show that they aren’t).
The total area of the smaller squares is simple. Each square has an area of $s^2$, so the total area is $a^2 + b^2$. For the larger square, the side length is $(a+b)$, so the area is $(a+b)^2$. But what is this result when you square it? Expanding the term simply gives $a^2 + 2ab + b^2$. This is awfully close to $a^2 + b^2$. In fact, the two expressions would match up if that pesky $2ab$ term wasn’t present. To do this, we would need $2ab=0$. But this implies that either $a=0$, or $b=0$ (or both). And that can’t happen in our situation, since we’re trying to find squares with the given property. Additionally, having a square of side length zero seems like we’re cheating, so we ignore that situation. Therefore, the answer is that we can’t find two squares of side lengths $a$ and $b$ which have a combined area equal to a square of side length $(a+b)$.
This is a nice result, because it turned a geometry problem (find the areas of squares) into an algebra problem. Most students will know how to expand $(a+b)^2$, but I’m guessing the geometrical connection might not seem so obvious. This puzzle teases out this precise notion. In fact, what it tells us is that we need to add a “correction factor” in order to make the equality hold. In this case, the correction factor is $2ab$, which we could also give a certain geometrical meaning.
I think that these kinds of puzzles are really what mathematics is about, fundamentally. It’s about these unexpected, surprising connections that can link different topics within mathematics to give answers to questions. I also find it challenges students to think about questions in different ways that aren’t necessarily obvious, which is a good skill to have.