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ZENODO
Preprint . 2026
License: CC BY
Data sources: ZENODO
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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Quantum Mechanics and Writing the LHS of dp/dt = Force in terms of a Gradient?

Authors: Ruggeri, Francesco R.;

Quantum Mechanics and Writing the LHS of dp/dt = Force in terms of a Gradient?

Abstract

In (1), it is argued that the Schrodinger (nonrelativistic) quantum equation may be developed from steady state type classical equations linked to the continuity equation and Newton’s second law. Given the nonrelativistic treatment of (1), F=dp/dt = d/dt (mv) = m dv/dt partial + mv dv/dx. In (1) an equation (III.2) for many particles is obtained from the equation for a single particle: Md (dv/dt partial + v dot grad v) = dF + hbarhbar/4m grad dot (grad grad d - 1/d (grad d) (grad d) ). In (1), it is further assumed that v = grad (phi) for conservative forces, where phi is a function of x,t. One may then see that for d=1 case, for a single particle, that phi= -Et+px and that one may write a Schrodinger equation which is essentially a conservation of energy equation. Here, we consider starting with dp/dt=Force and using a special relativistic treatment. We note that for a conservative force, F = -grad V and suggest trying to write the LHS in terms of a gradient for a free particle so the gradient may be “pulled out” from both sides of dp/dt=F. From a physical point of view, F is a pull or push, but we argue that V (potential) is also physical and that given that p is almost like a force, that there should be meaning to p= grad phi(x,t). This is the basis premise. In the case, F=0, we show that one may obtain -EE + p dot p = constant via this approach which involves writing p = d/dx (-Et+px) (one dimension). We note that associating dp/dt = F with a Lorentz invariant -Et+px shows the close association of Newton’s second law with special relativity because dp/dt=Force is relativistic as long as p=mov/sqrt(1-vv/cc). We suggest that this approach associates operators id/dt partial and -i grad partial (hbar=1) with E and p and allows one to write a free particle equation (Klein-Gordon) in terms of exp(-iEt+ipx), which is a quantum form. We suggest that one way in which to see free particle quantum mechanics emerge (albeit in the simplest case-that of a free relativistic particle) is to try to consider dp/dt in terms of a gradient suggesting that there may be x, t features even for E,p constant. In Newtonian mechanics, a constant E and p immediately implies that derivatives are 0. This need not be the case if one considers p = d/dx (-Et+px). The legitimacy of considering p (which is closely linked to a force) is linked to the legitimacy of considering F = -grad V. In other words, V is a potential which is another way of considering the features of a force and p = d/dx (-Et+px) may mean that -Et+px (and exp(-iEt+ipx) may be another way of considering p,E at a deeper level than simply stating that one has a constant E,p for a free particle. We argue that one does not need to consider a continuity equation as done in (1) and that the full approach follows from analysis of a single free particle (constant E,p) in the special relativistic case, which then yields also the nonrelativistic one.

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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!
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