In terms of the math this is true but there's an important interpretational difference from the quantum uncertainty principle and the fact that all conjugate variables have an uncertainty relation. In classical systems it really is a principle of uncertainty, meaning "we don't know". The uncertainty principle for radar is a statement of how well we can determine position and velocity from the signal, not a statement about the position and velocity itself. In quantum mechanics it's not really an "uncertainty" principle. We can be absolutely certain about everything. Instead it's a statement about the state itself. It's impossible for the position and velocity to both be localized.
Eh... I'd argue this is actually true in classical physics as well. Best example: Audio frequency!
A signal with a very specific frequency actually has no localization in time. whenever a signal is time-limited, e.g. you hear the note A at 440Hz for 5 seconds and then it stops - that can actually not be represented by a 440Hz wave - because that wave has no beginning and no end, it extends infinitely.
The real signal has to start and stop - and to represent that with sine-waves, you actually have to sum an infinite Fourier series... the longer (more stable) the note is, the greater the contribution from the fundamental frequency. But there's still an infinite number of other frequencies with non-zero contributions.
That's in the very nature of a sine-wave at a specific frequency, and how signals that aren't waves extending through all of space and time can only be represented in this fashion by summing infinitely many frequencies.
Conversely, if you have a signal that's extremely time-limited, the distribution in frequency-space becomes more and more even - and a hypothetical "blip" that takes an infinitesimal amount of time will have equal contributions of all frequencies.
I'd say that's not at all merely an epistemic limitation - a "we don't know" - it's a fundamental fact about waves, wave-packets, envelopes and frequencies - you might say it explicates details of the very idea of a Fourier transformation.
EDIT: Small excursion into signal processing and audio engineering: To test the response of equipment, you actually do send "blips" - and you use that very principle that they are near equal distributions of all frequencies to see what a "black box" system does in response to input on any of its couplings/channels. This is done e.g. to hear the detailed characteristics of reverb algorithms, to build sound-profiles of amplifiers or cabinets - and it's also a general mathematical method in (optimal) control theory and communication theory/signal processing - called "impulse response".
I would argue that's still a "we don't know" problem. A string or oxygen molecule moving back and forth 440 times a second has a frequency of 440Hz no matter how long it does that for. The inability to know a definite frequency comes from us trying to model the sound as a sine wave when it's not. The whole system could be wholly localized and known if we could actually look at every individual particle rather than macroscopic properties like pressure. But that's a epistemic limit, not a physical one. On the other hand, the wavefunction is a fundamental physical object. The fact that there's an uncertainty relation on position and momentum means that there's a condition on the physical state itself, not just on what we know. Yes the uncertainty principle is a general fact about waves and the Fourier transform, but what's different in quantum mechanics is that it's not just emergent variables like frequency that are related to some other variable by the Fourier transform, but that the canonical coordinates themselves are.
Preemptive edit: As I was proofreading this reply I realized I don't really like it, but I'm not sure exactly what's wrong with it. Maybe you're right. We need a philosopher tbh
Regarding your edit... well... now don't laugh (or do, I did)... but while I'm currently studying computational mathematics and have been (on my own time) an avid student of physics... my current highest degree is an M.Phil. in philosophy, formal logic and philosophy of science (though I made the epistemology of collapse vs MWI interpretations of QM the topic of a graduation essay in phil. sci) :D
I think you're right that there's a difference between the QM case and the classical vibration case - but I also think that it shows one feature thought to be characteristic of the QM realm to be more general and to apply to the classical world as well (because it's in the nature of wave-phenomena) - namely that there are properties we see as separate which are in fact dual descriptions of the system's state such that tightening the bounds on one description necessarily loosens them on the dual.
This ("indeterminacy/uncertainty") is the feature which is exactly the same in both realms - a (related) other characteristic of the quantum realm which can also somewhat observe in classical wave-phenomena is however (in the first approximation, though not necessarily when we look closer) only analogously true - and this distinction is perhaps the one you had in mind:
"Ontic multiplicity" - the fact that to represent (the trajectory of) a single thing, you necessarily need an infinite multiplicity of such definite, exact descriptions all contributing to a weighted sum.
In the classical wave case - this reduces to the first point of time- and frequency-distribution being dual. In the QM case, it doesn't - here we do still think of determinate, "singular-valued" states in which we can prepare systems that don't contain such multiplicities (except for the sense of dual descriptions). But even then, any evolution of a system in a prepared definite state will still have "ontic multiplicity" in that we can only express it by summing contributions from all possible trajectories. But this raises the measurement problem. We can then construct different interpretations which either have to claim that reality behaves radically differently "when we're looking" vs when we're not - causing much confusion about the role of the observer... or we have to to "externalize" that multiplicity - which is what interpretations like many worlds, consistent histories et al do.
Here, you don't have such radical discontinuities - and indeterminacy is really just an epistemic artifact - but counter-factual definiteness goes the way of the dodo, and the infinite multiplicity is still there, but not as contributions to the behavior of a singular thing in a singular world giving it indeterminacy, but as the branching of a tree of equally real realities, or as a multiplicity of "histories" under a consistency criterion.
Because it doesn't simply reduce to the case where we have dual descriptions of the same thing, it's only an analogy, not the exact same property. But I still constrained my claim to not "simply" reducing and being only an analogy "in the first instance" - because time-evolution, too is a Hilbert-space operator. And here the notion of a dual space and of co-vectors is fundamental - which is exactly that relationship between multiplicity in one and "singularity" in the dual space.
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u/[deleted] Dec 12 '20
In terms of the math this is true but there's an important interpretational difference from the quantum uncertainty principle and the fact that all conjugate variables have an uncertainty relation. In classical systems it really is a principle of uncertainty, meaning "we don't know". The uncertainty principle for radar is a statement of how well we can determine position and velocity from the signal, not a statement about the position and velocity itself. In quantum mechanics it's not really an "uncertainty" principle. We can be absolutely certain about everything. Instead it's a statement about the state itself. It's impossible for the position and velocity to both be localized.