Here is an excellent explanation of uncertainty principle by Richard Muller, Professor of Physics, U. Calif. Berkeley, coFounder of Berkeley Earth.
“Fourier analysis has a very important theorem: If a wave consists only of a short pulse, such that most of it is located in a small region Δx (read that as “delta x”), then to describe it in terms of sines and cosines will take many different wavelengths. The wavelengths, in math, are commonly described by a number k. This number is such that k/2π is the number of full waves (full cycles) that fit into 1 meter. Physicists call k the spatial frequency or wave number. A wave confined to a region Δx in size must contain a range of different spatial frequencies, Δk . Then the Fourier math theorem says that these two ranges have the following relationship: ΔxΔk≥1/2
“This equation has nothing to do with quantum behavior; it is a result of calculus. This theorem predated Heisenberg; Jean-Baptiste Joseph Fourier died in 1830. It’s just math—the math of waves, water waves, sound waves, light waves, earthquake waves, waves along ropes and piano wires, waves in plasmas and crystals. It is true for all of them.
“In quantum physics, the momentum of a wave is Planck’s constant h divided by the wavelength. The wavelength is 2π/k. That means we can write the momentum (traditionally designated by the letter p) as p=(h/2π)k=ℏk. Taking differences for two values of p, this equation becomes Δp=ℏΔk.
“If we multiply the Fourier analysis equation ΔxΔk≥½ by ℏ, we get ℏΔxΔk≥ℏ/2. Then we substitute Δp=ℏΔk to get ΔxΔp≥ℏ/2
“This is Heisenberg’s famous uncertainty principle. Once we accept that all particles move like waves, the uncertainty principle is a mathematical consequence.
“In math, the theorem wasn’t an uncertainty principle; rather it described the range of wave frequencies in a short pulse. But in quantum physics, the range of frequencies translates into an uncertainty of momentum; the width of the pulse becomes an uncertainty of where the particle will be detected. That’s because of the Copenhagen probability interpretation of the wave function. If different momenta (velocities) and different positions are available in the wave function, then making a measurement (such as observing it being deflected in a magnetic field) means picking one out, choosing one value out of many.”
The key point is that the Heisenberg’s uncertainty principle is the result of math applied to wave-like property.
The mathematical analysis assumes that we can always approximate a location in space as a Euclidean point. Is this assumption correct?
I believe that this uncertainty comes from using Euclidean point for location in space. The dimensionless Euclidean point seems to adequately define locations occupied in space by mass. But this doesn’t seem to be so for locations in space occupied by energy field.
When we look closely we find that location in space has an innate dimension equal to the wavelength of disturbance at that location. The de Broglie wavelength for matter is very, very small and it is possible to approximate it by the “dimensionless” Euclidean point. But the wavelengths of disturbance in energy field are not small enough to be so approximated.
The uncertainty seems to come from the use of this assumption for locations in energy field. We cannot “pinpoint” a location that has a significant dimension.
If this is properly understood and we can correct the mathematics being applied at quantum levels, then probably it will lead to much simpler understanding of the quantum phenomenon.
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As we discuss this and as we work our way through these phenomena, it seems intuitive to me that precise whole ratios, straight lines, square corners, and “Euclidean” space, etc., are abstractions only. They are useful, yes, “locally” but when looking closely, there seems to be no Natural phenomena that precisely represents these concepts.
This is one reason for the Uncertainty.
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Then uncertainty is in math that is being used. Isn’t it?
Can there be better math to define space and locations in it?
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I think of maths as language. I think of scientists as the explorers, historians, poets, and storytellers of the Universe. Maths are the languages and dialects of science. I think that before the eloquent maths come the inspiration.
But an illiterate like me knows just enough to understand what an ignorant boob that I am. I only understand the least little bit about about what has already been well worked out by men in science.
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Maybe a Newton will come along and see into the earlier Universe, and see clearly how it was that spacetime was once a unity, see into the roiling, seething processes that paradoxically both crush and rip apart spacetime.
It is a lot to bite.
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Yes, it is a lot to bite.
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. . . even if taken is little bites.
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But I think my point is at that point of inspiration, possibly that Newton will also have to invent the next dialect of mathematics to describe what he sees.
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New math is needed to describe space.
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My study of the interface between Physics and Metaphysics has, at this moment, led me to studying the interface between Mass and Field.
The concept of electromagnetic waves is vital to understanding of the field concept. Our idea of waves comes from waves on the surface of water, or sound waves in air. But unlike water waves or sound waves, the electromagnetic waves do not travel in a medium. They are their own medium. So math applied to electromagnetic waves have to be interpreted differently from the math applied to matter-centric idea of waves.
Electromagnetic waves are waves of space, and we don’t quite know what space is. Space splits into electrical and magnetic fields when it is disturbed. All we then have is a fluctuation of these fields. These electrical and magnetic fields are not really propagating in space. They are merely creating a fluctuating condition of space.
Math describes a pattern. That pattern is interpreted in a certain way for matter-centric waves. The same math pattern may apply to electromagnetic waves, but it would have to be interpreted very differently because there is no distance is being traveled. The distance is being “created” by the Electromagnetic wave so to speak.
So, the ‘x’ in the wave functions for electromagnetic waves means something entirely different. The ‘x’ applied to the disturbance in the nucleus is extremely small. It seems to overlap itself. What does that mean? If it is the wavelength then what does this wavelength mean within a nucleus? I now have to study wave equation without assuming that ‘x’ means distance.
It seems that we see distance from the perspective of matter. If there were no matter we shall see no distance. The whole idea of distance changes when only field is there. The field has extent, but it seems to be entirely different kind of extent. For example, what is the extent when the frequency of electromagnetic wave is very small and there is no mass around? We cannot say if the wavelength is very large or small because there is nothing to compare it to except to other wavelengths.
How do we compare two different wavelengths when there is no concept of distance? We compare distance by looking at the dimensions provided by matter. The dimension of matter is made up of infinity of infinitesimal wavelengths, and that is the basis of Calculus.
There is nothing wrong with calculus. We just need to make its interpretation consistent not only with matter but also with field.
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