The Heisenberg Uncertainty Principle
A fundamental aspect of quantum mechanics as misunderstood as it is popular.
While most avoid the deep waters of quantum mechanics, even those swimming around in the shallows have heard of the Heisenberg Uncertainty Principle (HUP). It’s usually expressed as the inability to know both a particle’s speed and position. Which is true but only a small, and slightly inaccurate, part of the whole picture.
One slight inaccuracy is the word “speed” — it should be the word “momentum”, but “speed” is more accessible. Everyone has a sense of what “speed” means. Not everyone has a good sense of what “momentum” is.
Momentum is the preferred physics term for objects in motion because it includes the notion of speed, the notion of direction, and the notion of the object’s mass. Momentum is the given speed in a given direction (the velocity) of an object with a given mass.
Another slight inaccuracy lies in the phrase “inability to know” — which suffers from ambiguity and imprecision. The accurate, albeit verbose, way to put it is that we cannot know to arbitrary precision both the momentum and the position of a quantum object.
The corollary is that we can know either one to arbitrary precision (which just means ‘as best as any instrument we can make will do’). And we can know both to some non-precise degree, depending on the system in question.
Further, this only applies to — and is a fundamental characteristic of — quantum systems. In the classical everyday world we inhabit, we can know both the momentum and position of objects just fine.
Another aspect of this is that, if we measure a particle’s position and then its momentum, we’ll get different results compared to measuring those the other way around. Measuring a quantum property disturbs the system (“collapses” its wavefunction), so subsequent measurements of other properties get different results than had they been done first. Physicists say such properties do not commute.
Speaking of measuring, this inability to know exactly where a particle is and where it’s going is not a limit of our instruments. Even theoretically perfect instruments cannot resolve this quantum uncertainty — it is fundamental to the fabric of reality.
Which tells us that, on some level, reality is fuzzy.
Wheeler’s 20 Questions Game
John Wheeler (1911-2008) is one of modern history’s greatest theoretical physicists and educators. He popularized the term “black hole” and coined such well-known terms as “wormhole”, “quantum foam”, and “it from bit”.
Relevant here is his version of the famous 20 Questions party game.
For those still in Plato’s (or other) cave, the game involves a “Guesser” who leaves the room while the “Hosts” decide on a secret object. It can be anything: President Lincoln; the first John Wick movie; a pepperoni pizza; an elephant; a pine tree; whatever. The Guesser returns and asks the Hosts up to 20 yes/no questions (“Is it an animal?” “Is it red?”). If the Guesser can guess the object with 20 questions or less, they win.
In Wheeler’s version, the Hosts do not pick an object. Instead, they answer randomly, except that each answer must agree with all previous answers. For example, suppose a question is: “Is it a living thing?” If the answer is “Yes” then all further answers must agree with the object being a living thing.
Eventually, the Guesser may feel they’ve narrowed things down enough to make a guess — which if they’re wrong (“Is it the Lone Ranger?” “No.”) counts as one of their 20 questions. It would be up to the Hosts whether to agree with the guess or not, although at some point the number of possibilities gets pretty narrow.
Which is taking the metaphor beyond its breaking point. Wheeler’s point is that reality apparently acts like his version of 20 Questions. The exact facts of the matter, at the quantum level, are not determined until we “ask” about them.
The game illustrates another fundamental fuzziness of reality not directly related to the HUP. The fuzziness here is related to quantum superposition, which I’ve mentioned only briefly so far. I’ll return to it in future posts.
Getting back to Heisenberg’s Principle, I turn to music and sound.
Either Or, Not Both
Imagine a song. Could be any song. Pick your favorite. (I’m thinking Slow Dance by Ana Popović.)
Now there are two questions we could ask of a song (two measurements we could take). The first is: “What is the melody?” The second is: “What note is it?”
Taking the song as a whole, the first question is easy to answer. The melody is the sequence of main notes — what we sing in the shower. But the second question doesn’t make much sense. The apparent answer is lots of different ones. It’s not a question that can be answered in the context of the song.
However, if we focus on any one moment of the song, the second question becomes easy to answer. The note is the note playing right here right now. But now the first question slips out of sight and cannot be answered. A note is not a melody.
So, whether a question makes sense depends on context.
In this case, both questions can be answered exactly, in any order, so the analogy breaks down there. The key point is that different questions can look at something from different points of view.
Here’s a more technical analogy that’s similar but which (because it’s more technical) bridges us towards a clearer understanding of the HUP.
Imagine a pure sound — a musical note of a single pitch with no overtones — a sine wave with a specific frequency. Again, we can ask two questions. The first is: “What is the frequency?” The second is: “What is the energy level?”
The first question requires looking at as much of the sound over time as possible. The longer we listen, the more accurately we can determine the pitch (because tiny variations average out over time). The second question, though, can only have an average because a sound wave varies its energy level (just as water waves vary their height so a large area only has an average height). Or we might say that the answer is many different levels (just as the song had many notes).
On the other hand, if we narrow our focus to a single spot on the wave, we can easily answer the energy question precisely. But now it’s impossible to say what the frequency of that wave is. A single energy level is not a wave. (A single data point is not a trend.)
The Fourier Transform
Interconnected properties that are incompatible, like pitch and melody, are often related through a Fourier transformation (most just call it a transform).
If you’ve ever seen a sound system with a spectrum analyzer, you’ve seen a Fourier transform. While the sound waves have an energy level that varies over time, a spectrum analyzer displays energy level over frequency. In the second example above with the sound waves, the two properties, frequency and energy level at a given point, are Fourier transforms of each other.
This gets technical and detailed (and a bit math-y), so I’ll leave it here. If there’s an interest, I can get into it another time. There’s a pretty cool simple geometric way to understand Fourier transforms. (If interested in more now, see this YouTube video or this blog post of mine.)
The key point is that there are complimentary pairs of properties that, even in our classical world, are mutually incompatible. In quantum mechanics, these incompatible properties become significant and form fundamental limits on what we can know about a quantum system.
Since quantum mechanics is a wave-based theory, Fourier transforms are relevant, useful, and important. The bottom line here is that position and momentum are Fourier transforms of each other. They are as incompatible as a note and a melody. Or more precisely, a frequency and an energy level.
Reality is Fuzzy
I’ve known even college physics professors to claim that the HUP is due to limitations in our ability to measure things. In their defense, even Heisenberg framed his explanations in terms of such:
At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum. [...] At the instant at which the position of the electron is known, its momentum therefore can be known only up to magnitudes which correspond to that discontinuous change; thus, the more precisely the position is determined, the less precisely the momentum is known, and conversely. (Heisenberg, 1927)
But these are just manifestations of the HUP in action. In reality, the incompatibility is fundamental to the math and, as far as we know, fundamental to fuzzy reality.
This fuzziness comes from two sources, firstly, from the HUP disallowing perfect information about non-commuting pairs (position/momentum, time/energy), and secondly, from the indeterminacy illustrated in Wheeler’s 20 Questions game, which I’ll write about another time.
That’s all for this time. Stay curious!
Do you know where the principle comes from? As in, how was it derived/discovered?