The Heisenberg Uncertainty Principle, Part 2

Welcome to part 2 of my explanation of the Heisenberg uncertainty principle. Today, I'm going to introduce the concepts of a quantum "state", an "ensemble" of states, and probability density. Only with this framework can you accurately describe what the uncertainty principle. But first, a video. This is from a TV show that tried to popularize mathematics, by showing its uses. Being employed as a mathematician, a lot of their explanations bugged me. This one especially bugs me.

Why? Read on.



Let's begin with a quantum state. Recall from my initial description of quantum mechanics that you can only have a guess at what a quantum particle is doing. This is what the quantum state describes. A quantum state contains all the information that you need to guess the properties of a quantum particle. Say you want to say what your best guess is for where an atom is located if you try to measure it. This information is contained in the quantum state.

This all sounds very abstract, but you deal with stuff like this every day, you just don't think of it this way. Let's say you just called your friend in the morning and asked him where he is. He says that he is at home. You have information about his state, and you can make an estimate about where he is. You don't know where in his house he is, but you have a guess. A state is a simple idea, it's a collection of information that you can use to make guesses.

A quantum state is denoted by something called a "ket". Written with something inside brackets like this |state information>. For our friend who is at home, you know he is in the "home" state, which you could express as | home >.  In quantum mechanics, we do this, but the information is usually more abstract. I'll go into much more detail about quantum states when we go through Schrodinger's equation later on.  For now, this is all you need to know.

The word ensemble means (in our context) a collection of states that are initially completely identical. For example. Imagine I'm holding an atom, and I set it on a table. Now imagine I have a bunch of other identical atoms, and I set them each upon a different, but identical table. This is an ensemble of quantum states. Simple.

The final concept I need to introduce is probability density. Say you have glasses, and that things are very blurry without them. You take a white piece of paper and draw the following.

Pretty boring. It's a dot, it has a defined size, and you know exactly where it starts and ends. You take off your glasses, and what you see is this.

Do you know where it is? You can guess, but it's distorted and blurred.  The center appears to be in the same place, but you can't say for sure. You are essentially seeing a probability density for where the dot could be, and your brain can decide where it most likely is. One number used to define the uncertainty, or width of this probability density is called the standard deviation. The wider the distribution, the larger the standard deviation, and the less sure you are of where the dot it.

As we discussed in the previous post, position and momentum are important. Each one of these has a probability density, and therefore has a standard deviation. And remember those delta x, and delta p things in the uncertainty principle? Those are simply the standard deviations (widths) of the probability density function!

Okay, so where does this arise? Let's look back at the atoms we had sitting on the table. Initially we knew where each of them were. Remember though, in quantum mechanics, we can on make a guess after that. I go to a table and look for the atom, I find that it's not where it started. Imagine that at that same moment, I decided to check up on all the other atoms. Some of them are very near where I started, but some are pretty far. If I place a dot at where I measured them, it looks something like this.

Now, if I have a bunch of these states (a large ensemble), eventually I get a picture that looks like this.

This is the probability density of this ensemble of states. What does this tell us? If I just look at one atom, it's most likely to be right where it started, but it's possible for it to be away from where it started. The darker the area, the more likely it is to be there.

Alright, let's apply this to the uncertainty principle. We have an ensemble of states and make a bunch of measurements of position and momentum. Say this time, we the uncertainty in position is small, that means the uncertainty in momentum is large. After performing the same measurement on a bunch of different states in the ensemble, we get these pictures.

If we had a good idea of the momentum, this picture would be opposite.

So the uncertainty principle tells us the limit of certainty to which we can know about a system. I recommend playing with the link I posted in the previous post, as you now know what a probability density is! So let's go review. Why does that video bug me? What's wrong about this statement? Well, he says that the act of measuring changes what the particle is doing. As if by performing the measurement bumps into a particle and moves it to a different place, or changes is momentum. While this does happen, this has absolutely NOTHING to do with the uncertainty principle. When you make a measurement, the particle assumes a particular position and momentum. The only way you can see the uncertainty principle come about is through repeated measurements of identically prepared states. The uncertainty principle is meaningless (but not any less true) for one measurement.  I would like your input and questions about this!

I know this topic is pretty dry, but this is essential for understanding quantum mechanics. Trust me, it gets way more interesting. For my next post, I'm going to take a small break in discussing what quantum mechanics is, and go to a REALLY cool prediction of quantum mechanics. It's called the Casimir effect, and is currently being researched. It's related to the uncertainty principle, and is one of the more strange predictions of quantum mechanics. After this slight detour, I will talk about Schrodinger's equation, arguably the most important equation in quantum mechanics. This equation gives us a method to calculate what the quantum state of a quantum particle is. Using the state computed by this equation, we can derive the probability density for position and momentum. Believe it or not, there are many useful and cool things this can get us. Stay tuned, thanks for sticking with me through this complicated subject!