You are thinking along a line that I am not in sync with. I am unsure how Fourier Analysis comes to play in your line of thinking. Along the line of energy being a function of wave-length (or frequency, same thing just inverse), Fourier Analysis doesn't come into play. So I am having problem following your line of thought here.

Simply highlighting a fallacy of analysis -- at the simplest level, wave-particle duality is identical to time-frequency duality, and the Heisenberg uncertainty principle is simply the relationship between time-domain bounds and frequency-domain bounds.

That is to say -- the universe is more than happy to allow conditions which, within the scope of a Fourier analysis, should count as a photon (say) with a frequency lower than the age of the universe -- that you consider it as such, is merely your fault of applying an analysis that can only resolve things in terms of frequency, and not in other, more suitable terms.

(What those terms are, is an exercise for the student, naturally...)

Now, I'm not sure under what conditions you could ever observe such a thing --

-- but the takeaway point is, use what analysis is most suitable; Fourier analysis falls apart at "DC", where "DC" is merely however long you wish to look at a signal.

Remember also that Fourier analysis (of our simplest, most favorite functions) is symmetrical: the transform must exist

*for all time, including all negative time and all positive time*. Neither condition of which can be properly met in a finite-time universe!

Fortunately, Fourier transforms fail softly, so we can dirty up our graphs by bounding them within realistic windows. We remind ourselves of the extents and limitations of our experiments, and perhaps we choose to exclude that pesky DC term from our subsequent analysis because it's an artifact of the transformation, or measurement. But remembering, also, that we should contemplate its origin, in case it's really there (the universe has a net charge..?!).

In QM, Fourier isn't

*quite* right, because QM isn't pure signal analysis. But the duality phenomenon is common to all wave systems, and so we should naturally expect to see similar concerns arise in all wave systems.

Basically, for QM, you might find it's better to use a time-domain analysis than a frequency-domain analysis, in such a case. The frequency-domain (or momentum, or..) results arise from eigenvalues of the solved equation; the eigenfunctions give their spacial distribution. This works nicely when the problem is static (like the energy levels of a particle in a box, or the hydrogen atom), for which you expect a frequency analysis to work nicely (because it's not otherwise changing over time!).

The choice of analysis, is a convenience to the solver -- consider solving for the time-domain waveforms of an RLC circuit (analytically, not with SPICE

), versus with Fourier analysis. Once you've trudged through all the awful integrals and found your series of exponential functions, you still can't do much with it because if you want to change the input signal, you have to integrate the damn thing again (output signal = convolution of input signal with impulse response).

AC steady state analysis is doing the whole thing in the Fourier domain, though they don't often tell you that that's what you're doing (hey, it's only the second course in the average EE curriculum).

If nothing's changing over time, of course the two approaches converge; we don't even bother writing the integrals nor the reactances, and the whole thing reduces to DC resistor networks: EE101.

Conversely, Fourier analysis won't help you much with a switching supply circuit -- it's bad enough if the duty cycle is varying over time, but if the frequency is varying as well, you're pretty much screwed.

Combined with the nonlinear parameters in a real semiconductor circuit, you're better off using energy arguments and events in time.

I am also unsure of your example Earth-Sun system analogy in reference to the very low energy discussion.

Do you scale Planck's constant with it?

Nope, as stock. Basically take the already-solved hydrogen atom equations, and plop in the correct potential (gravitational vs. Coulomb) and masses.

The issue is not the absolute size of the quantum, rather, the issue is how close is it to Planck's constant. The closer to Planck's constant, the bigger the uncertainty. At the size of the solar system, uncertainty due to the uncertainty principle is not even in the scale of rounding errors. So even if you are talking about a single particle of graviton, you are talking a huge amount of energy far exceed the scale of uncertainly. There would be no chance of it hiding within the grey area covered by the uncertainty principle.

That said, much much much bigger "borrowing from uncertainty" came into play before - namely the big bang.

The nice thing about this example is, it illustrates the problem of limited analysis over a much more human time scale than the Big Bang. Since, as you say, the math doesn't care, you can simply plug in any number -- why not ask the same questions of an atom 2 A.U. across, or 100pm across?

For the Earth-Sun system, some pertinent questions are:

- It is well studied that the Earth's orbit affects the orbits of its neighbors. If it radiates so little energy, how can this be? (Even given the Earth has been in its orbit for 4.5 billion years.) Surely, so little energy cannot distort space-time enough to do that!

- If radiation is given off at the rate it seems to be given off at (and, well, why wouldn't it?), then how is it that we can seemingly measure its effect on a far shorter time scale? (The answer to this, on the truly quantum scale, is a field of active development, actually -- the mechanism is analogous to using parametric sensors to measure the presence of a signal, altering the signal slightly in the process but not absorbing it whole.)

If you don't know/remember the QM to work it out by hand (it's a good study to work through, I encourage you to give it a try if you can!), the answer is here.

SPOILER:

http://www.physicspages.com/2013/01/15/earth-sun-system-as-a-quantum-atom/Tim