Implicit Constructs In Wave-Particle Duality

[petrushkagoogol]

In wave-particle duality, is the particle the time differentiation of energy, and the wave the time integration (continuous) of mass across the median distribution of it's loci ?   

[exchemist]

In wave-particle duality, is the particle the time differentiation of energy, and the wave the time integration (continuous) of mass across the median distribution of it's loci ?   

No. The rate of change of energy with time of a QM entity in a stable state (e.g. an electron bound in an atom) has a value of zero! 

 

And the integral of mass over space simply gives the mass of the entity, which has a constant value.

 

It make no sense to try to say a particle or a wave or any physical entity "is" a numerical value! 

 

The entity will have properties, and these properties will have values. A QM entity such as an electron has some fixed properties (mass and charge and intrinsic angular momentum or "spin"), plus a load of  other properties such as position, momentum, kinetic energy and potential energy, which depend on the environment it is in. None of these "is" the electron. All of these are properties pertaining to what could be thought of as a particle. But it also has a wavelength and a frequency. 

 

It is not possible in QM to separate out what is the "particle" and what is the "wave": the two behaviours are inextricably combined. 

 

The best way to attempt to grasp the link is to think (a) of de Broglie's relation between momentum (p) and wavelength: λ=h/p.   (h is Planck's constant). The more momentum a QM entity has, the shorter is its wavelength (and the higher the frequency of the wave); 

 

and ( b ) to be aware that waves can be superposed on one another in a way that can reinforce or reduce the amplitude. 

 

With these two concepts in mind it is possible to understand that the information one has about a QM entity determines how particle-like or wavelike it may appear, in a given context.  If the momentum is well-defined, the wave is well defined, but the amplitude of the wave will extend over a wide area of space, so you will have little information on its location. So it is not very particle-like: it is smeared out over space. If, by contrast, its location is well-defined, the only way a wave can have its amplitude concentrated in one spot is if you make a superposition of lots of waves, all of different wavelength, but with one common, central, maximum peak.  Then the differing wavelengths will lead to progressively more and more destructive interference as distance extends from that central point, giving you a nice well-defined location. But then you do not have a well-defined wave: you have a superposition of a whole lot of different ones! Which means its momentum is not well-defiined. 

 

This is the basis (in wave rather than matrix form) of Heisenberg's Uncertainty Principle, in its momentum:position form. Δp.Δx >/= h/4π. (The standard deviation imprecision in momentum multiplied by the imprecision in position is equal or greater than Planck's constant divided by 4 pi.) There is an animation of the principle of superposition here: https://en.wikipedia.org/wiki/Uncertainty_principle#/media/File:Sequential_superposition_of_plane_waves.gif

 

There is also another aspect of the uncertainty principle, which does involve energy and time, though not in the way you suggest. It is also the case that ΔE.Δt >/= h/4π, in which ΔE is uncertainty in the energy of a QM state while Δt, (not entirely intuitively, it has to be said) is the lifetime of the state. The effect is that It is impossible to know the energy of a very short-lived state accurately.  This is observed in "pressure broadening" of spectral lines, in which at higher gas pressures, collisions between the emitting species become so frequent that they shorten the lifetime and thus make the energy of the emitting state less well determined, leading to emitting over a wider range of frequencies.  

 

But this is digressing so I shall stop here. 

Edited April 23, 2018 by exchemist