The Heisenberg Uncertainty Principle, Part 1

Hold on readers, these next couple posts are going to be rough. The Heisenberg Uncertainty principle is fundamental to understanding quantum theory, but is a rather confusing topic. It isn't extremely complicated, but it is VERY subtle and easily misunderstood. Through the next couple posts, I hope to illuminate this highly misunderstood topic. Let us begin this venture with a short clip from one of my favorite television shows.

Let's break down this video. The two horses are racing and when it comes down to the end, the horses are so close that they are both within a tiny tiny distance each other. In order to measure which one is closer, the judges must use an electron microscope to tell the difference. Professor Farnsworth (if you aren't a fan of the show, he's the scientist at the end..) exclaims, "No fair! You changed the outcome by measuring it!"  This is one of the odd things about quantum mechanics, and it's related to the uncertainty principle. The principles of quantum mechanics, as discussed briefly in the previous post, say that you don't have any certainty about an object's properties; very specifically, this means its position, and its momentum. If you do not know what momentum is, don't be alarmed, I will explain. Momentum is formally defined as the product (multiplication) of an objects mass (essentially it's weight on earth), with its velocity (it's speed). What does this mean? Say you got a train, and it's moving at 50 miles per hour, let's say it weighs 50 tons (I don't know, I'm pulling numbers out of the air...). This means it's momentum is 50 mph * 50 tons, or, 2500 mph*tons. Don't worry about the units, if mph*tons is confusing, ignore it. Now imagine you have a car going 50 mph, but it's much lighter, say it weights 2 tons. It's momentum is only 100 mph*tons. Both are going the same speed, but the train has much more momentum.

Why do I bring this up? As it turns out, position and momentum are the most fundamental quantities, and any other measurement of interest can be expressed using both position and momentum and nothing else. Cool, huh? Understanding what exactly momentum is isn't critical for this discussion, and for purposes of this post, you can think of position as simply where something is, and momentum as a rough measure of speed. It should come as no surprise that the uncertainty principle is written in terms of these two quantities. Without further ado, here is the Heisenberg uncertainty principle expressed using.... dun dun duh! Math.

Do not be alarmed, I will break down this expression in simple terms. Let's start on the left, and make our way to the right of the expression. First you see this triangle thingy. That's the Greek letter delta. In physics, it represents a change or in this case, the uncertainty. You can pretty much read the first part as uncertainty in x. x represents the object's position. The next part is delta p, where the p stands for the object's momentum. The symbol to the right is read "greater than or equal to." Essentially, the values an the left are AT LEAST as big as the values on the right. The funny looking h is called h-bar, it's is a fundamental value of nature. It's physical significance is actually very elegant, but that involves some VERY complicated mathematics.

So how do we read this equation? The product of the uncertainty in position times the uncertainty in momentum is greater than or equal to some number (a very small number).  Now you're probably thinking, "...okay, Dan, so I multiply two things and it's bigger than something small, who cares?" Well consider this. I make a measurement of the position of a small objects, say, an atom. I decide that I'm VERY VERY sure about where it is, does this make the uncertainty large or small? Small. Very small in fact. Here's where it gets interesting. The smaller my uncertainty in position is, the bigger the uncertainty in momentum is, pretty weird, huh? The more sure I am about where the atom is, the less I can be sure of how fast it's going. This is roughly what the uncertainty principle tells us. Truth is, this introduction was rather sloppy.

I hope this introduction made sense, I will be revisiting this topic in the next post in more exact terms. I am hesitant to post this now, but I will anyway. Here is a Java applet that shows the relationship between position and momentum in turns of their "probability density function." The concept isn't too bad, but if it sounds intimidating, don't worry, I will be describing this in detail later, and it will be hopefully made crystal clear. Just open up that applet if you are feeling adventurous and try messing with the width of the density function.

There is an interesting note about the uncertainty principle that has nothing to do with quantum mechanics. As it turns out, the uncertainty principle is also extremely important (read, problematic)  in the field of signal processing. Instead of being a relation between position and momentum, it's a relation between time and frequency. My job is in signal processing, but I study quantum mechanics and relativity in school, and it's interesting to see connections between the two fields. As it turns out, there are quite a few connections between the two that I will discuss, stay tuned!

Sorry there were no cartoons this time, there will be some in the next post! Please feel free to ask questions or post comments, just be sure to keep civil! If something is confusing, please let me know.