Relativity Calculator
"The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts." - Bertrand Russell ( 1872 - 1970 )
• The subatomic world of nature at its essentially reduced smallest is comprised of localized packets of energy best shown as waves comprised of a variety of wavelengths.
• Because these localized packets of energy are comprised of a variety of wavelengths of different waves, therefore a wave-particle also possesses a variety of momenta since
• We can now plainly see that in order to maintain constancy in the above equation, that as the composite variety of wave lengths for a mass-particle becomes overall shorter, the overall composite magnitude of the variety of momenta must become greater. This can also be seen in the equation
where the shorter the wavelength λ of the mass-particle, the higher will be its frequency and hence carry a greater amount of energy which translates into a larger amount of momentum.
That is, as composite wave length λ becomes shorter, all of the composite characteristics of the wave-particle - frequency, energy and hence momentum - will gain unit magnitudes in their respective probability distributions. And the opposite will also be true as well, that is, as the composite wave length of a wave-particle becomes longer, therefore the other composite characteristics of frequency, energy and momentum will also all decline in their respective unit magnitudes and will do so as a probabilistic distribution.
Hence, whenever position of a mass-particle of wavelength λ is more accurately determined owing to a given opening in a wave-diffraction experiment, the less accurate will momentum be determined. And, the opposite is also true - namely, that as momentum of a mass-particle of wavelength is more accurately pre-determined, the exact location position will be less accurately identified in any wave-diffraction experiment.
For example, if we shoot a given mass-particle thru a slit opening which defines a given allowable wavelength λ to pass
we also discover a probability pattern of strikes or "spreading out" ( distribution of intensity ) on an observation screen of this wave diffraction phenomenon for the mass-particle.
Now, for slit openings considerably greater than the mass-particle's wavelength λ, the mass-particle propagates thru more nearly in a straight line of passage and can be observed as such. However when the slit opening is more nearly constrained to the theoretical wavelength λ of the mass-particle, all of the wave characteristics of the mass-particle become apparent as shown here:
Finally, as the slit opening is adjusted for greater or lesser composite wavelengths λ of the mass-particle to pass, we discover that there will always be an lower bound for the conjugate of the composites of wavelengths and momenta as follows:
And in terms of overall position and momentum, the probability distribution density curves for intensities of momentum will again arrive as:
Hence the uncertainties in either the location and/or the momentum of a mass-particle in terms of its wave function at any given instant must be at least equal to or greater than h, Planck's Constant.
Therefore, if for example we attempt to "pin down" or locate a mass-particle, Heisenberg's calculation will give us higher and higher velocities ( remember: momentum = mass x velocity ) for more and more precise locations. On the other hand, if we somehow slow down the mass-particle or even achieve a precise velocity ( or momentum ) for the mass-particle, the location "spreads out" and becomes fuzzy and hence probabilistic!
Oh, also notice that Heisenberg's Uncertainty Principle has absolutely nothing to do with any interference with the mass-particle by the observer's measurement technique or instruments involved. It rather solely pertains to nature's inherent wave function for sub-atomic mass-particles and E=mc2 from which p = E/c!
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