Powered by OpenAIRE graph
Found an issue? Give us feedback
image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ ZENODOarrow_drop_down
image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
ZENODO
Preprint . 2025
License: CC BY
Data sources: ZENODO
ZENODO
Preprint . 2025
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2025
License: CC BY
Data sources: Datacite
versions View all 2 versions
addClaim

Quantum Wavepackets, Bohemian Potential and Interaction Probability Part 2

Authors: Ruggeri, Francesco R.;

Quantum Wavepackets, Bohemian Potential and Interaction Probability Part 2

Abstract

 In Part 1, we argued that the Heisenberg uncertainty principle (standard deviation p) * (standard deviation x) >= hbar/2 only applies when a measurement (i.e. interaction) has occurred. Otherwise, the notion of standard deviation does not exist. Thus, if one has a Gaussian wavepacket, it is the measuring device which has created it and constrained is standard deviations of momentum and position. A free particle, which has not been measured, is described by exp(-iEt+ipx), as argued in Part 1, and not by a wavepacket. Here we ask: How does one interpret a wavepacket calculation? In (1), we argued that exp(-iEt) is a free particle energy conserving probability and exp(ipx), a momentum conserving one. Then, a sum of exp(ipx - iEt) represents an OR situation in probability theory. There is superposition and so one obtains a weighting of possible outcomes which is the probabilistic interpretation of the scenario. If one considers two slit interference, the viewpoint is the same. exp(ipx) is a probability which allows for momentum hits of p in a range of dx=hbar/p outside the center-of-mass, which follows x=vt. This built-in stochasticity of impulse delivery leads to the notion that probabilistically, a particle may interact with both slits if their separation is about hbar/p. One writes an OR probability statement: W(x)= exp(i p dot r1) + exp(i p dot r2), where r1 is the position vector from the center of slit 1 to a point far away on a screen, and r2, a vector from the center of slit 2 to the same point. The result, W*(x)W(x) is a probabilistic expression which describes the possible outcomes probabilistically. (1). Seen in this way, the Gaussian wavepacket should be interpreted in a similar manner and not as a physical description of the particle which moves freely. A single bound state in a potential V(x) is localized by a potential and one may consider only OR cases of exp(ipx), i.e W(x) = Sum over p a(p)exp(ipx). In the measurement case, there is no quasi-permanent localization and one uses a linear superposition of exp(-iEt+ipx)s. Nevertheless for a Gaussian wavepacket, two probability distributions describing possible outcomes are created. One has W*(x)W(x)= exp(- (x-u)(x-u) / (2 sigma(t) sigma(t))) where sigma is the x standard deviation. In addition, W(x) = Sum over p a(p) exp(ipx), leading to a momentum outcome distribution of a*(p)a(p). As written, sigma(t) for x which means that the Gaussian is expanding. We called this unphysical if one considers the wavefunction as representing a description of a single physical entity (2) at times beyond t=0. Like in the 2-slit case, we argue that it represents an outcome distribution. Thus, one would not consider any more time values than t=0, at which time the measurement is completed. An exp(-iEt+ipx), describing a free particle, is measured by a specific device which creates the Gaussian form and gives it a specific standard deviation in space and momentum. The measurement is then done and W*(x,t=0)W(x, t=0) represents the possible outcome positions of the particle at t=0 (i.e. probabilities for x portions of a phase shift). The momentum distribution yields possible p outcome values. As soon as the measurement is finished, the particle moves as exp(-iEt+ipx), with p being probabilistically linked to a*(p)a(p) (a Gaussian as well), where W(x,t) = Sum over p a(p)exp(-iEt+ipx).If one has a second measuring device which creates a tiny momentum standard deviation, one would be able to carry out experiments which would determine roughly the standard deviation of p created by the first measuring device. Furthermore, one would be able to measure phase shifts as well by comparing a free particle measured by a device with a tiny p standard deviation and a large x one with free particles which were not measured.

  • BIP!
    Impact byBIP!
    selected citations
    These citations are derived from selected sources.
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    0
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
Powered by OpenAIRE graph
Found an issue? Give us feedback
selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
Green