Quantum Mechanics | Physics | Simplified
1 — Ehrenfest’s Theorem: Explained
An introduction to quantum operators and commutators
To do justice to Ehrenfest’s theorem, I will split it over two articles. In part one, which is this article, we’ll explore quantum operators and commutators. Right now this may sound weird and mathsy but like with everything else I’ve written, all you’ll require is a high school understanding of math and the rest will follow. In part two, we’ll establish what expectation values are put together our understanding of Ehrenfest’s Theorem.
The (generalized) Ehrenfest’s Theorem can be thought of as a bridge between classical and quantum mechanics. Classical mechanics, of course, is everything that was postulated before the advent of relativity. And quantum mechanics is the weird, strange stuff that makes you question your sanity.
Some Groundwork
It would be a foolish attempt to explain this equation without laying some groundwork. The first piece of information necessary to understand Ehrenfest’s Theorem is the function of (quantum) operators.
If we happen to be studying a quantum system, then an operator is a mathematical object that we apply to that system. Or more specifically, apply to the wave function of that system. I realize that all I've done so far is throw a fair amount of jargon at you. So let’s break down what I’ve just said.
First, wavefunctions. What are they? Briefly explained, a wavefunction is a mathematical description of everything we know about a particular quantum system. A quantum system is of course a “system” we happen to be studying using quantum mechanics. It could be anything. Put an electron in an electric field and that’s a system. And if we use quantum mechanics to study it and it becomes a quantum system. In fact, the wave function is very, very, very closely linked to the probability distribution of a system. Probability distribution is a “density” function, f(x). While the probability is a specific value realized between one and zero. Well our wave function, when squared, tells us how likely we are to find a particular particle at different regions of space in our system.
|Ψ⟩² ∝ probability distribution of finding a particle at a particular point.
I’ve talked about this in a lot more detail in a previous article, so if you haven’t already, check it out:
Well, this brings us to operators. Like we’ve already established, operators are mathematical objects that we apply to the wave function of our quantum system. They basically “operate” on quantum states. They do things to the quantum state. To make things clearer, let’s take a particular example.
The Position Operator
This operator, evidently, is labeled with the letter x with a hat on top. Or in more specific terms, a circumflex. The hat on top of the operator differentiates it from other quantum mechanical entities. If you see a variable in a quantum mechanical equation with a hat over it, you can say, without batting an eye, that it must be an operator. Now if we were to make a measurement of where our electron is in space, then mathematically it is the equivalent of writing this:
We have applied the position operator to our wavefunction. This eventually gives us some information about the system but we won’t delve too deep into that. The first thing to consider is the notation. We’ve written our operator on the left-hand side of the wavefunction ket. I’ve talked about kets and bras in previous articles, like the one linked above, so check them out if you’d like. But every time we see a wavefunction and an operator on its left, we know that the operator has been applied to the wavefunction. The fact that the operator is written on the left side will become important shortly.
But at this part, we should really consider the fundamental difference between classical and quantum mechanics. Let’s say that our quantum system is a single electron. In our system, like the one we have below, a particle can move forwards or backward since it’s confined to one dimension. a single dimension.
Quantum mechanics tells us — or at least the Copenhagen interpretation tells us — that before we make a measurement in our quantum system, there’s some probability of finding our electron at every region in space. That means you could find the electron at r = 1, r = 3, r = 9, and so on… Before making the measurement, we can only work out the probability of finding our electron.
It’s this function like we established, that is the important one. Specifically, this function is telling us — Ψ² is telling us — that at r=2 because Ψ² is large, we’re most likely to find our electron. Whereas at 4, Ψ² is small and so we’re least likely to find our electron. Now there are a few subtleties to this but for our purpose, all we care about is that this Ψ² function is telling us the probability distribution of finding our electron in our one-dimensional system.
If we made a measurement and found our electron at r = 2, we’d witness the wavefunction collapsing from both ends and converging at 2. What we’d see is the probability density being zero at all other regions of space. Because before we made the measurement, there was some likelihood of finding our electron at all values of r. And after we made the measurement, we knew that our electron is in one specific place. Even if we only knew that for an infinitesimally short amount of time. But very quickly that probability distribution starts to spread out and so we become less and less sure of our measurement as time progresses. This is in accordance with Schrödinger's equation; which if you haven’t read it already, is linked down below.
Again there are obvious intricacies that I’ve left out but they aren’t necessarily important to understand operators and their function. The essential idea is that the wavefunction of the system changes and making a measurement is what causes that change. So making a measurement has genuine and evident consequences for our system. In a way that it doesn’t for classical mechanics.
Mathematically, making a measurement is written using an operator. We’ve talked about the position operator already but a couple more examples follow:
Now the Hamiltonian is the big guy. You’ll see this written everywhere, even in the Schrodinger equation. This operator is the total energy of the quantum system. For a single particle system, it accounts for kinetic and potential energy. The expression for a single-particle system’s Hamiltonian would look something like this:
However, for larger systems, we need to start considering electrostatic attractions and repulsions. For example, the generalized Hamiltonian for the Helium atom looks pretty daunting
We’ve seen that making a measurement results in the collapse of the wavefunction. But I believe it’s important to point out that it’s not the presence of a conscious being that causes this “collapse”. It’s the quantum system’s interaction with the instrument that is the result of this collapse. Anyway, back to operators.
Since we’ve established that making a measurement has an evident consequence on the system’s wavefunction, making two measurements is even more complicated. Because now, the order in which we make those measurements matters.
Going back to classical physics, let’s say we were to measure the position of a football on some arbitrary plane followed by its momentum. Well if the ball is stationary, i.e, not moving, then we could find it at any given coordinate on the x-y axis. Now for convenience’s sake, let’s say we happen to find it at the point where x = 5. Since it’s not moving, its velocity, v, is 0. And since momentum, p, is equal to the product of mass and velocity, its momentum too would be zero. So the result of our two measurements is as follows:
- Position → x = 5
- Momentum → p = 0
If we were to measure momentum and position instead (notice the order of measurements) then we’d still find the same results:
- Momentum → p = 0
- Position → x = 5
No big deal. The results remain the same regardless of the order in which we make the measurements. This is necessarily not the case with quantum mechanics. Since when a measurement is made on a quantum system, its wavefunction collapses. So when measuring the momentum after the wavefunction’s collapse, we’re actually measuring a completely different wavefunction. This may have sounded hazy but here’s a simpler understanding:
In other words, measuring the position first and then the momentum of our system can yield a different result when compared to measuring the momentum first and then the position. We know which operator is applied first based on where it is placed. The operator closest to the wave function is applied first (the one on the left-hand side). The left side applies the position operator followed by the momentum one. While the right side does it the other way around. Now because these two values are not equal, subtracting them will yield a non-zero result.
Ignore the right-hand side for now but from here, we can do a series of neat mathematic manipulations.
If we first start out by factorizing the left side, we get:
The quantity inside the brackets is known as the commutator between the two operators.
For notation’s sake, we write the commutator with square brackets around the two quantities:
The reason we define these commutators is so that we know whether the order in which these measurements are made matter or not. In this case, with position and momentum, the order does matter since the commutator has a non zero value:
Specifically for position and momentum, this is actually true when these quantities are measured in the same direction. Since momentum is a vector (has both magnitude and direction), it can be resolved into different components for each arbitrary direction.
When these quantities, position, and momentum, are measured along a particular axis — let’s say the x-axis — then they do not commute. In other words, if you measure the momentum component along the x-direction first, followed by the position, it would not result in the same value as when the position is measured first followed by the momentum.
In cases such as this, where the commutator is not equal to zero, we say that the operators do not commute. The order in which we make the measurements do matter.
Let’s take a more generic scenario. We’ll conjure up our own arbitrary operators: A and B. It doesn’t matter what they are. Here, we could have a scenario where these operators commute. Measuring A first makes no difference to the general result of our measurement. In other words:
In this particular case, we say that A and B do commute. In other words,
AB = BA
Order Matters
But coming back to the equations where these operators yield a non-zero value. This means that they’re equal to something but for the purposes of this article, we don’t need to necessarily understand what they’re equal to.
If operators commute:
you can, without batting an eyelid, say that the order of measurement does not matter.
If they do not commute:
Well, then the order matters.
Like always, there are mathematical intricacies that have been left out but I hope you have a general understanding of the language quantum mechanical systems are built on. I’ll end this here and the following article will explore quantum expectation values and we’ll finally put it all together to understand what Ehrenfest’s theorem is trying to tell us.
Thank you for reading!
