The Real Nature of the Heisenberg Uncertainty Principle

 The Real Nature of the Heisenberg Uncertainty Principle

By Frank Lin

 

The Heisenberg Uncertainty Principle has been a common example of the “mysticism” of quantum mechanics. It tells you that if you know the position of a particle, then you can’t know where it’s going. It may seem bizarre, but hey that’s just quantum for ya, where everyday logic breaks down! In fact, it even allows for brief violations of conservation of energy! Particles at the quantum scale don’t even have real positions or momentum. How cool is that? Well, there’s one problem. Everything that I just said is false. The Heisenberg uncertainty principle does not describe these things at all, and while what I said is the popular depiction of the principle, the popular depiction of the principle has been flawed over the decades. Every single “pop-science” video on the HUP (Heisenberg Uncertainty Principle) I’ve seen online have all been false. This leads to 2 puzzling questions.

1. What’s the right interpretation of the HUP?

2. How do so many educators get the HUP wrong?

We will be answering these two questions in this blogpost. So buckle up, it’s time for quantum.

So, what is the Heisenberg Uncertainty Principle?

The Heisenberg Uncertainty Principle is this equation right here 

I know, it looks like a bit much. This is the generalized uncertainty principle, which means its applies to many variables. Let's dissect it. 

Q and R are variables. They can represent things like position, energy, and momentum. That curly O you see there is a standard deviation (more on that later). Q hat and R hat represent the operators of Q and R respectively (operators are what we use to calculate quantities). Those brackets represent a commutator. The commutator between two elements A and B is defined as AB-BA. Of course, if AB=BA, then the commutator of A and B is zero. The commutator algebra helps us know whether two variables commute. 1/2i is just a constant that doesn't matter much for our purposes. When we plug in momentum and position in this equation, we get 

This is the most popular representation of the Heisenberg uncertainty principle. The two variables here are position and momentum. Now what does this actually mean. To know that, we need to talk about quantum systems. 

Imagine you have an infinite amount of identical quantum systems. Each system is in the same identical state. Behind each system is a physicist that is curious of measuring the position of the particle in the system. Now when the physicists measure the position of the particle in their system, you'd expect each one of them to get the same answer, since these systems are identical. But when you actually do this, you don't get the same result. The results are inconsistent. This is indeed puzzling--does it mean that the particle was in many positions at once? No. Does it mean that the particle doesn't have a real position? No. What this means is that you cannot measure the true position of a particle with confidence, since a single measurement on multiple identical systems will yield different results. Let's say each physicist plots their result on a graph. The spread of the results is the standard deviation. That is what the first symbol on the left hand side means. 

Now let's say these physicists also measure the momentum of their system. Like before, the results are inconsistent. Now here's the good part--if you modify the system in a way where the spread in position measurements is small, the spread of momentum must now be big. The reverse is also true. If the spread in position measurements is small (say an interval of 105.46 x 10^-27nm), that means that you are fairly "certain" about where the particle is, since it must be within that spread. And since the deviation in your position measurements are small, then the deviation in your momentum measurements must be larger than the deviation in position (according to the equation, it should be >0.5kg m/s. Still seems pretty precise, but it is orders of magnitude less than the position spread). We can entertain even smaller numbers of position (such as 1.6 x 10^-87) to get even bigger spreads in momentum (you can try this yourself by plugging it into the inequality). 

But wait, how does the equation (or more exact, the inequality) even imply the tradeoff here? Well, the equation says that the product of the position SD and the momentum SD must be greater than or equal to some constant value (h bar/ 2). This means that if one value goes down, the other must go up to stay just above (or equal to) the constant value. And that, is what the HUP really means. I hope I haven't made this too complicated, but often a significant level of complexity is needed to get the correct picture. Now, I have made a few simplifications in this whole text. Standard deviation doesn't exactly mean the spread in the measurements of a variable. The standard deviation is the range where at least 75% of the data points lie in twice that range. So in a sense, it is a way to measure the "spread" of data points. A narrow wave has a small SD. A broad one has a large SD.

In the beginning of the blog, I mentioned that the Heisenberg uncertainty principle applies to any two variables. We dealt with momentum and position, since those are the most common examples (the reason being that most quantities can be expressed as a function between position and momentum). Let's deal with another pair of variables--Energy and momentum. It turns out, that when you plug in energy and momentum in the generalized UP, you get zero. This is because the energy and momentum operators commute. It should be apparent to you guys that uncertainty stems from 2 operators not commuting. If 2 operators do commute, such as the energy and momentum operators, you get no uncertainty. Since the standard deviation in energy multiplied by the standard deviation in momentum must be greater than or equal to zero, that means that you can measure energy and momentum simultaneously with arbitrary accuracy. Well, it may be a stretch to say you can measure both with arbitrary accuracy. Remember, there is an uncertainty relation with momentum and position, so the SD of momentum cannot be zero at any time--since it would break the momentum-position uncertainty. So in better words, you can measure energy and momentum both with arbitrary accuracy except for perfect accuracy.

Sidenote**: You might be wondering how we can actually calculate the standard deviations for any 2 values. The process is very straightforward. The standard deviation of a variable is equal to the expectation value of the squared variable minus the expectation value of the variable squared, all under a square root.

There is another popular uncertainty relation called "Energy-time Uncertainty". It may be misleading to call it that, since it does not stem from the generalized uncertainty principle. There isn't even any time operator in quantum mechanics. It is a mere coincidence that the energy-time uncertainty inequality looks like the momentum-position inequality. Irregardless, (take that grammar nazis) we will still call it that since that is what it is commonly called. 

Ignore the delta signs and imagine standard deviation signs in their place. The second inequality is the time-energy uncertainty inequality. This one says that the standard deviation in the energy times the time it takes for the expectation value of energy to go up another standard deviaton is greater than or equal to the constant. In a different blogpost we will talk about this in more detail. The reason I add it here is that some people say this equation justifies the "creation" and "annihilation" of these things called virtual particles. This is a common misinterpretation of time-energy uncertainty. It speaks nothing about energy conservation. All it says is that the smaller the standard deviation of energy, the more time it takes for the expectation value of energy to go up one standard deviation. For now, expectation value just means "average energy" (this is a drastic simplification). As for whether or not virtual particles are real, well the short answer is no, but that's an in-depth question for next time. Now that we've covered what this relation actually means, why do people get it wrong so often?

Why do educators get the Heisenberg uncertainty principle so wrong?

I do not mean to insult the may educators that teach QM. By "educators" I am really referring to pop science YouTube videos and newspapers, as that is where the majority of the population gets their quantum mechanics info from. Actual professors get it right most of the time. This part of this blog is opinionated, so this is just my personal speculation. Quantum mechanics is a hard subject. Hard subjects are often misinterpreted, and when enough people who misinterpret QM spread their interpretation as fact, you get misconceptions, which are things widely thought to be true even when they aren't. Almost all things in quantum mechanics are misinterpreted, and the HUP is no exception. Having or spreading a misconception does not make you a bad person (we have all had them at one point or another). The good thing is that correcting your misconceptions open new doors for learning and understanding even more about the universe-- and what's more to physics than that?