The Magic of Monte Carlo – Part 1
posted by: Jamie Antonelli
Quantum mechanics, our best theory for describing the universe at very small scales (about the size of an atom or smaller), is the foundation of modern particle physics. At its heart, quantum mechanics is based on probabilities. This means we can’t make predictions with the same certainty as we can for normal, everyday things. Imagine a world where every time you let go of something, it had the same chance of floating upwards as it did falling downwards: welcome to the world of quantum mechanics.
So, in order to learn about physical systems that are very small, we need to make many measurements and average their results. Imagine you didn’t know anything about coins, and you want to study what happens when you flip one. So you flip it, and let’s say it comes up heads. You might think, “OK, problem solved, when you flip a coin, it always comes up heads!” (Remember, you don’t know anything about coins.) Let’s say you flip it again and get heads. You’d be even more sure you were right! But, on your third flip, it comes up tails… Now you’d have to rethink your solution. At this point, you would conclude that a coin comes up heads two thirds of the time and tails one third of the time. You still wouldn’t be right, but you’d be closer to the real answer. And the more times you flipped your coin, the more you’d close in on the correct answer of 50% heads and 50% tails.
Here is an illustration of the idea that with probabilities, the more data you collect, the better you understand what’s going on. I rolled a die 10,000 times, and make graphs of how many times i rolled each number, at different points in the data-taking.
Questions to the reader: How do the plots change going from left to right? What shape would we expect if we rolled the die like a bajillion times and what would that tell us about what happens when you roll a die?
Ok, so I didn’t actually roll a die 10,000 times… I used a computer to do it for me. That’s what I’ll talk about in my next post – simulating probabilities with random number generators.