Chapter 5: Uncertainty Principle

In our daily lives, our intuition is that information is cumulative. For example, suppose that I first observe a football and note that it’s blue. I then perform a subsequent measurement (using a tape measure) and find that the ball is 1.00 ± 0.01 meter to my right. Then, I know that the ball is both blue and 1.00 ± 0.01 meter to my right. 

However, what if gaining knowledge about the position of the ball affects its color? That is, just by measuring property A of an object, we will change property B of that object.

This effect goes by the name of the “general uncertainty principle” in quantum mechanics.

While this equation may look complicated, its meaning is quite simple to understand. First, ΔA is the uncertainty of the property A (and ΔB the uncertainty of property B). In other words, ΔA describes how well we can measure the property of an object. In the example above, the uncertainty of the football’s position is ΔA = 0.01 m — the ball is between 0.99 and 1.01 meters to my right.

The right hand side looks more complicated. We can ignore the factor of 1/4, i, and the left/right angle brackets. The important bit to focus on is: [A,B]. In mathematical lingo, this is called the “commutator of A and B”. Specifically, [A,B] = AB - BA.

AB means measure property B, then property A. And BA means measure property B then measure property A. If the order in which we measure properties A and B does not matter (and our classical intuition tells us that it shouldn’t!), then AB = BA and the value of [A,B] is zero. This means that 

Since the right hand side is zero, this means that the uncertainty of A and B can be as small as we want. Or in other words, there is no limit to how well we can measure both A and B. 

However, what does it mean if AB - BA is not zero? This means that by changing the order that you measure properties A and B, you change the results you get. And so, measuring A must change something about B, and vice versa.

In the quantum regime, this fact is not so surprising. For example, let’s say quantity A is the position of an electron, and quantity B is its velocity. To measure the position of an electron, we must look at it! For example, if we scatter (bounce) a photon off the electron the photon can impart some momentum on the electron and change its velocity.

The uncertainty principle is not a uniquely quantum phenomenon. It is simply the statement that when two measurements are incompatible (e.g. one changes another) then we are fundamentally limited in how well we can understand the value of both quantities. However, in our day to day lives, this fundamental limit is far below the precision with which we can measure quantities, and so we rarely notice it.

To conclude, we leave with some questions for the reader:

1. A radar speedometer measures the speed of an incoming car by emitting radio waves (RF photons) and observing the photons that bounce of the car. By how much does this measurement process change the velocity of the car?

2. Are the images below an example of the uncertainty principle?