Quote:
Originally Posted by BigDon
What are, if any, macro effects that can be directly attributed to this?
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Probably
none, with a sufficient constraint on what "macro" means. But the uncertainty principle in the sense of ΔEΔt≥h/2π (where "E" is energy and "t" is time) sets the minimum possible width for a spectral emission or absorption feature, and is measurable in a spectroscopy lab. That might not be "macro" enough for you, but I felt like mentioning it to point out that the uncertainty principle is not just the usually cited ΔxΔp≥h/2π but in fact applies to any conjugate pair of quantum mechanical variables, which includes energy and time.
Quote:
Originally Posted by Chris Hillman
For the mathematically inclined: someone should probably mention that the Uncertainty Principle can be considered to arise naturally in functional analysis ...
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Chris beat me to the point. I have long been fascinated by the amazing coincidence between the behavior of the physical universe (which we assume to exist independently of ourselves) and the mathematical universe (which is an invention entirely of our own minds). That the latter should be so good at predicting the behavior of the former is astounding to me. And a good example of this is the totally unintuitive uncertainty principle from quantum mechanics. In fact it can be derived purely from mathematics with no input from physics, or any consideration of the physical world at all, yet the physical world obeys the purely mathematical constraint.
There is a derivation from Fourier Analysis here:
Fourier Transforms and Uncertainty Relations (hosted by the ever useful
MathPages.com). This is also discussed in the book
From Classical to Quantum Mechanics by Giampiero Esposito, Giuseppe Marmo & George Sudarshan; Cambridge University Press 2004. The niche this book fills for me, which the other quantum mechanics books don't, is the extensive treatment of the methodological links between classical & quantum mechanics, so you can see how they blend together. See section 4.2, page 128, "Uncertainty relations for position and momentum". Here we find out where the uncertainty relation comes from (emphasis in the original):
"The uncertainty relations result directly from Fourier analysis and hence are not an exclusive property of quantum mechanics. A possible formulation is as follows: a non-vanishing function and its Fourier transform cannot both be localized with precision. Indeed, in the framework of classical physics, if f(t) represents the amplitude of a signal (e.g. an acoustic wave or an electromagnetic wave) at time t, its Fourier transform ft shows how f is constructed from sine waves of various frequencies. The uncertainty relation expresses a restriction with respect to the measurement in which the signal can be bounded in time and in frequency band."
From Classical to Quantum Mechanics, page 128.
So there is an implication that you could measure some fundamental uncertainty in any signal bandwidth, and that should count as macroscopic. But in the presence of more mundane & ordinary noise, I think it would be a challenging measurement to make, assuming that my interpretation is correct.