It’s a lot more than just being unable to measure two things simultaneously.
“In the future, maybe quantum mechanics will teach us something equally chilling about exactly how we exist from moment to moment of what we like to think of as time.” –Richard K. Morgan
If you want to know where something is, you just measure it to greater and greater accuracy. Rulers can give way to calipers, microscopes, and even individual particles of light of ever-shorter wavelength. Yet the more accurately you measure an objects position, the more inherently inaccurate your knowledge of its momentum becomes. It isn’t just a failure of our instrumentation; that uncertainty is fundamental to the Universe. Physically, this is known as the Heisenberg Uncertainty Principle. Where does it come from? That’s what Brian McClain wants to know:
Explain to me what information is gained from the quantum mechanical commutation relation. There’s more to it than, “we just can’t measure both properties at the same time.”
It’s true: you can’t measure both properties at the same time, and yes, there’s more to the story.
When you learned mathematics way back when, you probably heard of some properties: associative, distributive, and commutative, for example. The commutative property is the one where, for example, “3 + 4 = 4 + 3,” as in the example of addition, or “3 × 4 = 4 × 3,” for multiplication. In classical physics, all variables commute: it doesn’t matter whether you measure position and then momentum, or momentum and then position. You get the same answers either way. But in quantum physics, there’s an inherent uncertainty that arises, and measuring position and then momentum is fundamentally different from measuring momentum and then position.
If you want to know a particle’s position in one (say, the x) direction and its momentum in the same direction, there’s a difference in what you get depending on your order-of-operations. What the quantum mechanical commutation relation says is that if you do “position and then momentum” versus “momentum and then position,” the two answers will be different by exactly the amount iℏ, where i is the square root of (-1), and ℏ is the reduced Planck’s constant. It works this way for position and momentum because they’re the Fourier transforms of one another.
When you take this quantitative relation into account, you discover that there is a physical uncertainty that comes out. But it isn’t an uncertainty in measuring both variables together, but in eachvariable. In particular, what you learn is that you always have an uncertainty in position (Δx), and you always have an uncertainty in momentum (Δp), no matter how accurately you measure either one. Moreover, the product of those uncertainties (ΔxΔp) must always be greater than or equal to ℏ/2. It’s impossible to know any quantity that obeys this quantum relation to an arbitrary accuracy.
It isn’t restricted to position and momentum, either. There are plenty of physical quantities out there — often for esoteric reasons in quantum physics — that have that same uncertainty relation between them. This happens for every pair of conjugate variables we have, just like position and momentum are. They include:
- Energy (ΔE) and time (Δt),
- Electric potential, or voltage (Δφ) and free electric charge (Δq),
- Angular momentum (ΔL) and orientation, or angular position (Δθ),
along with many others. That last one, though, is particularly interesting.
Imagine you have a particle, and you know, inherent to that particle itself, that its intrinsic angular momentum (or spin) is ℏ/2, which is exactly the case for an electron. You decide to measure its spin in one particular direction, perhaps by passing it through a specially crafted magnetic field. The particles either deflect upwards (if their spin is +ℏ/2) or downwards (if it’s -ℏ/2), with no other possibilities. Therefore, you reason, I’ve determined these orientations very well.
It’s true: if you took all of those spin +ℏ/2 particles and passed them through another, identical magnet, they’d all deflect upwards. But if you rotated your magnet, to a perpendicular direction, the information in that direction was completely destroyed by that first measurement, so they can split left (for +ℏ/2) or right (for -ℏ/2) with a 50/50 probability. What’s worse? If you then took the results of either of those further-split ones and passed them through another magnet with the original orientation, they’d split again, +ℏ/2 and -ℏ/2, in the upwards and downwards directions.
In other words, when you minimize the uncertainty in one variable, you maximize the uncertainty in its conjugate variable. The existence of that uncertainty, the amount/magnitude of that uncertainty, and which variables that uncertainty occurs between, is what the quantum mechanical commutation relation tells you. And this is not without its extreme usefulness! You can derive the size and stability of atoms — why an electron never sits atop the nucleus in an atom — from this relationship. You can derive wave-particle duality and quantum confinement from this. And, remarkably, from the magnetism and angular momentum example, you can develop Magnetic Resonance Imaging (MRI).
It’s true! While a properly-configured magnet will cause a particle to split dependent on its angular momentum, a magnetic field that changes with time in the right way will force a particle into a certain spin configuration. These time-varying fields causes a quantum system to oscillate between those two states, and that is the “resonance” in Magnetic Resonance Imaging. The same principle is also at play in atomic clocks, in hydrogen masers (which are microwave-frequency lasers), and the hyperfine splitting of atomic transitions. Not bad for a simple relation that says, “AB is not equal to BA” for the right quantum setup. There’s a lot more than “we can’t measure both properties at the same time,” in fact, there’s a whole modern, quantum Universe to discover as a result!
Ethan Siegel is the author of Beyond the Galaxy and Treknology. You can pre-order his third book, currently in development: the Encyclopaedia Cosmologica.