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A bit more:
The difference is that in the many-peaked example by the pier, we are dealing with an assumedly infinitely long wave. Like a sine wave or a cosine wave. These are periodic, ever-repeating, infinitely long. They have perfectly well defined frequency / wavelength... but they have no central location on an infinite number line. They are so symmetrical that they defy a locality of position.
With infinite precision of the wavelength, we have lost all precision of the "central" location of the wave.
With a wave pulse, we can demonstrate mathematically that this is, in fact, a superposition of many sine and cosine waves. In order to have a finite number of peaks, it takes an infinite number of individual sine and/or cosine waves which compose it.
This one's a bit tricky to visualize, but I know you play guitar. Tuning your strings, you listen for beats. This is because you hear 2 notes at the same time, and the physics says that adding two waves is equivalent to multiplying to other waves.
So the waves you pluck have wavelengths A and B. You generally tune to unison, so these are nearly identical pitches.
Ignore the sloppy math here, but
sin(A) + sin(B) = 2*sin( (A + B)/2 )*cos( (A - B)/2 )
So the first term is the average of A and B. If you pluck the first string, and the second string (voicing them in unison), you hear the average of the pitches... with beats. The second term is the beats. It is 1/2 the difference of A and B. As you tune A and B to unison, the term inside the cosine goes to 0, and cos(0) = 1. So when you no longer hear the beats, you know that A = B.
I.e. they are in tune. The wave in the sin term reduces to sin(A) which is equal to sin(B), and the cos term multiplies this by 1.
The 2 out front means that the wave has 2 times the intensity, which we hear as ( sqrt(2) = ) 41% louder.
I say all of this to convince you that the math justifies treating a single wave (the multiplied version) as an additive combination of other waves (the + version). I hope to have given you a personal experience that I have explained with math, which also jives with you intuitively.
In the guitar tuning, we did not make a wave pulse, but a standing wave. An audible wave pulse sounds like a drum with no ring to it, a sudden rapport with no tone, a snap, bang, pop, or thud.
The more atonal, the fewer wave pulses.
The more tonal, the more pulses.
(Knocked it out of the park, here... This is your ears and brain dealing with uncertainty in counting the number of peaks.)
So I propose to you that we examine the wave pulse again. Recall, this represents a particle with a well-defined position.
This time, we don't want to eliminate the beats. We want a clever series of beats that has exactly one peak and is everywhere else 0. However, we are left with no option than to use sines and cosines to try and do so. Nature is a cold, rude and uncompromising meanie-face sometimes.
So... if I start with one wave. I can add another of twice the frequency, which cancels out every other peak. Then I can add another wave with a frequency that cancels out the next set. Then I do all of those. Then I realize that my solutions may have created their own peaks. Refine and do it again.
Fortunately for us, the Fourier Transforms does all of this in one step.
Oh crap... That means the position and momentum are a Fourier Transform pair. No matter how localized the results are in one of them, the results are equally de-localized in the other... Tiny variance in one = high variance in the other...
delta_x >= {ongbonga}/delta_p
delta_x*delta_p >= {ongbonga}
It just so happens that with solutions to the Schroedinger Equation in this universe,
{ongbonga} = hbar/2
where hbar is h/(2*Pi). Where h = Energy/frequency of a photon.
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Originally Posted by wufwugy
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