Non-commuting operators

The title of this post is a technical term that describes another perplexing aspect of quantum theory. In everyday life we get used to the idea that physical objects have very definite characteristics. Your laptop has a definite width, position, weight etc, and it has all these properties at the same time. According to quantum theory, some characteristics of particles are fundamentally incompatible with each other. An electron, for example, can't have a definite energy and a definite location at the same time.
Again, there is nothing in everyday life that is like this implication of quantum theory. The best we can do is to give a rough idea of what is happening.
Suppose you had a particular sum of money. It is possible to express that sum of money in many different ways which are financially exactly the same but physically different. For example, you could express it in dollars or in euros or in pounds, and whatever currency it was in, you could have it in different combinations of notes and coins.
In real life you can have your money in US coins, for example, and you could take one of the coins- say a quarter- and measure both its width and its thickness. Quantum theory doesn't allow that. It is as if quantum theory says that if you want to measure coin widths then your money will always be in US coins, but if you want to measure coin thicknesses your money will always be in UK coins. So let's suppose you start with a UK penny and measure it's thickness. When you try to measure its width it changes to a US quarter and no longer has the thickness you measured when it was a penny. And having measured the width of the US quarter, when you try to measure its thickness it turns into a UK two pound coin and no longer has the width you've just measured, and so on. In this crazy analogy, you can pin down one of the dimensions at a time, but when you try to pin down the next the previous one changes.
Quantum theory says that electron energy and position are just like that. When the electron has a definite energy it no longer has a definite position, and when it has a definite position it no longer has a definite energy.  However, perhaps the most amazing aspect of quantum theory is the way in which it predicts probabilities of experimental measurements of such incompatible quantities. We will look at this in the next post.