The Dan Feldman Wavefunction

Greetings, everyone! During my lunch break today, I was thinking about quantum mechanics (that’s what everyone thinks about while eating, right?), and started to wonder what my wave function might look like for any given day. I’ve mentioned wave functions before (see: Zeno’s Paradox), but I’ll give you a quick refresher before delving into my thought experiment.

In quantum mechanics, we usually talk about wave functions when describing the probability amplitude of the position(s) of one or more particles/atoms/molecules in space. In simpler words, it’s the likelihood of finding a particle somewhere in space at a given time. Wave functions have high values where the probability is high, and low where you are unlikely to find a particle. Before I give you an example of a particle’s wave function, let’s take a classical physics (large) example—me.

In classical physics (and our everyday lives), objects have one position at a given time. I’m currently sitting at my desk, and twenty minutes ago, I was in the bathroom. If you were to try and find me right now (i.e. measure my position), there is a certainty that you’d find me at my desk. If you were to try and find me twenty minutes ago, there’s a certainty that you’d find me on the toilet (and annoyed, you creep). The world is deterministic, because we can find where something is at a given time, and it follows the laws of physics.

Lets say you performed this experiment over and over again—10 times, to be exact—trying to find me during my Thursday office hours. You find me at my desk 9 times, and once on the toilet. Then my measured probability amplitude would have a value of 0.9 at my desk (I was there 90% of the time) and 0.1 at the toilet (I was there 10% of the time). For those who are interested in the mathematical form of your findings, this would be represented by two delta functions. The probability amplitude is really the square of the wave function (represented by a greek Psi squared |ψ|^2). Just some math, not that important to this entry.

But what if I was a quantum system? In quantum mechanics, wave functions technically represent the same thing—probability amplitude of your position. But it’s a little weird. Think about what I told you about my wave function. According to your experiment, you find me in one of two places…at my desk, and on the toilet. Classically (and via common sense), we know that I’m really in one place or the other at any given time…it’s just that if you do the experiment over and over, you sometimes yield different results. But I only occupy one place (or state) at a time. But in quantum mechanics, we don’t interpret this the same way. If I was a quantum system, I would be in BOTH places at the same time, but then when you look at one place (say you walk into my office), there is a 90% probability I will be there, and upon seeing me (making a measurement), my state is altered to be 100% at my desk. Weird right?

So I got to thinking…on any given day, where am I most likely to be? I’m a man of habit, and I spend most of my time in the same couple of places. So, I calculated what my wave function might look like. And here’s the results (click to enlarge):

If anyone can come up with a mathematical representation of this–100 points.

For those uninitiated, my office (and classes) are on the 5th floor of the College of Arts and Sciences (CAS) Building. Not the most exciting life, but I’m a first year graduate student at the moment, so it’s not all that surprising. Classically, the way you interpret this graph is thus: If you want to find me, there is a certain probability that I will be in those particular places…I’m only in one of those places at any given time, but if you were to try and find me every day, these are how many times you would find me in said places.

Quantum mechanically, as long as you haven’t gone looking for me yet (measured my position), I could be, and AM (sort of) in all of those places at the same time. I’m in what’s called a superposition of all of these states at once, and once you measure my position, I become in only one of those states. And if you do this over and over, I will be in one of those states a certain amount of times based on the probabilities in the graph. This notion is a bit absurd, and leads to the paradoxical thought experiment known as Schrödinger’s Cat (but that’s another story).

Nobody really understands quantum mechanics, or at least what the true physical nature of these wave functions is, but this is one interpretation usually accepted by physicists. It’s worth noting that in physics/chemistry, we can calculate some truly awesome looking wave functions for particles, atoms, and molecules…it’s pretty wild.

What does your wave function look like?