Heisenberg’s (and the general) Uncertainty Principle
An introduction to uncertainty principles (no math, I swear)
Uncertainty principles go beyond position and velocity— the story told today perhaps does not do justice to the importance of it all. The general attitude adopted towards science is to rush through the origin to get on with the actual science. But to do science, one must understand how it came to be. The idea of Heisenberg’s uncertainty principle is incomplete when just looking at our 20th-century quantum mechanical triumphs. The actual principle is, of course, rather more involved — to the point where oversimplification verges on disingenuous.
In essence, uncertainty principles are foundational concepts within Fourier analysis, suggesting that a nonzero function and its Fourier transform cannot be precisely localized. To elaborate, localization refers to how a function behaves near and away from zero. For the nerds, any mathematical assertion that combines a function describing localization with its Fourier transform can be considered an uncertainty principle.
However, it’s important to note that the uncertainty principle, in its various forms, is not originally tied to quantum phenomena. It’s not as grand as we make it out to be. It’s just math, really — the math of waves, of water waves, shockwaves, sound waves, microwaves, electromagnetic waves, waves along ropes, and those in plasmas and crystals. Just waves. The principle predates Heisenberg by about a century and finds its home in Fourier’s notebook.
Fourier analysis presents a crucial theorem: If a wave primarily consists of a short pulse, tightly concentrated within a small region Δx, expressing it using sine and cosine functions requires considering multiple different wavelengths. In mathematical terms, these wavelengths are often characterized by a value denoted as k, the wavenumber. It signifies that k/2π represents the number of full cycles contained within a wavelength. When a wave is confined to a region Δx, it must encompass a range of diverse spatial frequencies, Δk. This is where the uncertainty principle joins the party, stating that these two ranges relate as ΔxΔk≥1/2.
Mathematically, this theorem describes the spectrum of wave frequencies present in a short pulse, and it isn’t inherently an uncertainty principle. However, in the realm of quantum physics, this frequency range translates into an uncertainty in momentum. The width of the pulse becomes a measure of uncertainty in the particle’s potential detection location. This interpretation arises from the Copenhagen interpretation of the wave function. Because position and momentum spaces are Fourier transforms of each other, precision in one domain implies less precision in the other.
Let’s consider the classic example of a Gaussian distribution representing a particle’s position. The narrower the Gaussian, the more precisely we know its position. However, when you perform a Fourier transform on a Gaussian, it results in another Gaussian with a width inversely proportional to the original Gaussian’s width. In simpler terms, as the position uncertainty diminishes, the momentum uncertainty increases. In the extreme case of an infinitely narrow Gaussian in position space, you end up with a delta function (indicating absolute certainty) in position space, which transforms into a complex exponential (indicating absolute uncertainty) in momentum space.
Clickbait again.
It’s been a while since I wrote a mathy article. Bear with me here. What follows is a very standard derivation. There are equal ways to get here without invoking Dirac algebra but I find it far more convenient, especially since we are dealing with the abstraction of vectors.
For some observable, Q, we know that the variance, σ², is the expectation value of the difference between each expected value and the mean, squared. Written down, it’s actually pretty neat.
Thankfully, the expectation value can be rewritten as an inner product of the wave function and the corresponding operator on the same wave function. Mathematically, this yields
Here, abusing the hermiticity of our operator Q-⟨Q⟩ (since Q is Hermitian), we can take it off the second member of our inner product and slap it onto the left side. An important reminder for those of you who’ve forgotten your linear algebra: a Hermitian matrix is one that is equal to its conjugate transpose. This property allows us to mess around within our inner product.
Π denotes a new quantity |Π⟩ = (Q-|Q⟩)Ψ. Now, we invoke the Schwarz inequality. Note that the inner product of two vectors is expressed as
we can deduce that the norm of the inner product of a, b squared must be less than or equal to the product of the magnitudes of these vectors.
The proof is trivial (not in the way Fermat meant it). And I know it’s monstrous but I will leave this as an exercise to the reader. If you need a place to begin, let some new vector |c⟩ equal the following
and use the fact that ⟨c|c⟩ ≥ 0. Anyway, back to our variances. Invoking the Schwarz inequality for two observables, Q and R, we conclude
Since we know that
we can substitute ⟨Π|Φ⟩ = z to derive the following because the inner product is often complex anyway.
This looks pretty daunting but our equations are shaping up to resemble an uncertainty principle of some sort — we already have the variances on the left and the inequality’s in place. Naturally, the next step would be to simplify the right-hand side. Physically, it’s quite difficult to develop any motivation of what it means but let’s unfold everything and let the dust settle.
Expanding out, we are left with
And since the expectation value of some observable, O, in the state |Ψ⟩ is
we can get rid of our Ψs and write the equation as
Similarly,
It’s fairly straightforward from here. We’ve only got a little more to chip off. We substitute these in the parent equation.
And if you know about commutators (which I hope you do since you’ve made it this far), you’ll recognize the right-hand side neatly gives us
Now, we really let the dust settle.
This is the generalized uncertainty principle in its glory. Viscerally, it may seem like nonsense. Even having squared it, the denominator renders the right side negative and since you’re taking the product of two variances (both positive and real), you can’t permit such a violation. That’s what first comes to mind. It’s worth noting, however, that i is actually intended; the commutator carries an i of its own and they cancel out quite nicely.
Since Q and R are arbitrary observables, they serve little purpose besides abstraction. The goal is to get our hands dirty. So, consider (as Heisenberg’s uncertainty principle does) position and momentum.
We determine the commutator [x, p] with some arbitrary function, f.
Substituting this gives us our desired result.
This proves Heisenberg’s uncertainty principle but now it’s easy to see how this is the result of a far more general theorem. There will be a similar result for any pair of observables that have incompatible operators. Evidently, then, they do not share a set of eigenvectors either.
This article has been a lot of \langle and \rangle spamming. Excuse me if something is off somewhere but please feel free to let me know in the comments. As always, thank you for reading! Have a great day.
