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Planck's quantum constant


Heisenberg's Uncertainty Principle

"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 )


This can also be seen in the equation

Planck's quantum constant

where the shorter the wavelength Planck's quantum constant 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 Planck's quantum constant 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 Planck's quantum constant 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 Planck's quantum constant to pass,

werner heisenberg's uncertainty principle experiment

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 Planck's quantum constant, 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 Planck's quantum constant of the mass - particle, all of the wave characteristics of the mass - particle become apparent as shown here: 

Single-slit diffraction ripple tank

Finally, as the slit opening is adjusted for greater or lesser composite wavelengths Planck's quantum constant 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:

wave-diffraction equation analysis

And in terms of overall position and momentum, the probability distribution density curves for intensities of momentum will again arrive as: 

quantum wave probability densities

Dirac's Constant

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 Planck's quantum constant, 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 Planck's quantum constant  from which Planck's quantum constant.

Examples Using the Heisenberg Uncertainty Principle

Planck's quantum constant

The Large Hadron Collider ( LHC ) at CERN will be accelerating protons close to the speed of light, Planck's quantum constant, whose rest mass is

Planck's quantum constant

§ Case 1: Before achieving smashing protons at close to  Planck's quantum constant, let's suppose that the protons are speeding at  Planck's quantum constant with a 1% measurement precision or

Planck's quantum constant

Therefore, the uncertainty in measurement of proton velocity is

Planck's quantum constant

and by the Heisenberg Uncertainty Principle, the uncertainty in simultaneously determining proton velocity and position is given as follows:

Planck's quantum constant

§ Case 2: The LHC at CERN, however, will be attempting to smash protons at the speed of light, or at least virtually close to it! The uncertainty in proton position with the same 1% measuring precision as for proton velocity, becomes as follows:

Planck's quantum constant

§ Case 2a: Suppose that the smashing of protons at the speed of light does actually occur but only for a time duration of  Planck's quantum constant  with a 1% precision in time measurement.

Heisenberg's Uncertainty Principle tells us that

Planck's quantum constant

§ Case 2b: The rest energy, Planck's quantum constant, of the proton is

Planck's quantum constant

The amount of time that it would take to make this energy measurement to a precision of 1% is found similarly as before.

Planck's quantum constant

note : some of these examples are used in the future upcoming Relativity Science Calculator Mac application for Heisenberg Uncertainty Principle.

§ References:

  1. Planck's quantum constant "Equivalence of wave-particle duality to entropic uncertainty", by Patrick J. Coles, Jędrzej Kaniewski, and Stephanie Wehner, The Center for Quantum Technologies, National University of Singapore, Singapore, all of whose mathematical modeling conceptually combines Heisenberg's Uncertainty Principle with the Quantum duality of the wave - particle heuristic in addition to mathematically unifying 'wave' with 'particle' concepts.

  2. Planck's quantum constant "Simultaneous observation of the quantization and the interference pattern of a plasmonic near - field", by L. Piazza, et al., published March 2, 2015, setting forth a unique experiment making imagery, simultaneously, of the quantum wave and particle characteristics of light photons.

  3. Planck's quantum constant "Does the ψ-epistemic view solve the measurement problem?", by Shan Gao, Institute for the History of Natural Sciences, Chinese Academy of Sciences, September 6, 2015. In the problem of measuring features of a quantum ψ-wave function, do the pre-existing eigenvalues of the collapsing wave represent an updated amount of epistemic data of an underlying final reality and should therefore be accepted, then-and-there, at face value? And with that the quantum wave "measurement problem" is therefore solved? Or, should a deeper ψ-ontological ( -ontic ) model of reality be pursued to better encompass an admittedly rather superficial and constantly changing ( updating ) ψ-epistemic model? But, still, is the ψ-ontic model itself a lost fool's errand otherwise according to Heisenberg's 'Uncertainty Principle'?

    Thus in the physics quest for quantum reality, it therefore becomes in a final analysis a philosophic quest for ultimate reality.

Planck's quantum constant

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Planck's quantum constant

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