Time Base Force (Impulse) Versus Space Based Force (-dV/dx) in Classical and Quantum Mechanics
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2021
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/5137660 |
Daftar Isi:
- If one considers two colliding particles in a center-of-mass frame, each with different mass, then they have equal and opposite momentum or energy flow. After the collision, they also have equal and opposite momenta. A similar idea holds for a particle breaking, with force, into two pieces of different mass. It seems that a force, equal and opposite strength but existing only for a period of time, pushes the two unequal mass particles apart. Thus, time is broken into equal portions in an impulse type reaction. From this one may obtain Newton’s second law. Newton’s law, however, is usually applied to a space based force i.e. F(x)=-dV/dx, so the same law holds for impulse type forces as well as space based ones. In classical statistical mechanics with a potential V(x), one has both impulse (time-based forces) as well as a space-based one. The impulse (time-based force) creates the equilibrium. In quantum mechanics, one begins with a space based force F(x)=-dV/dx or V(x) a space based potential. We argue in this note that V(x) is actually linked to a time-based (impulse picture) in classical mechanics with the V(x) picture being of a statistical nature i.e. dp = Integral -dV/dx dx/v(x) where 1/v(x) (v(x)=velocity) is proportional to the spatial classical density. We also try to link the sharpness of V(x) to space resolution and momentum. In quantum mechanics, one tries to create a statistical theory based on free particles which undergo impulse hits or a time based force. Thus, a space based force (related to V(x)) is converted into a series of impulse (time-based force hits). Unlike classical statistical mechanics, one does not have the two separate kinds of forces (time and space). The time based (impulse hits) must create an average appearance of a space based force (-dV/dx). How may this be done while maintaining a momentum based space resolution? Impulse (time based force hits) occur anywhere in space so a kind of mapping of the time based impulse hit to space must be done. This may be done through exp(ikx) where k is the momentum hit delivered by a time-based force which has a spatial wavelength proportional to 1/k. Thus, different exp(ikx) with weights (which are actually time-based impulses acting at any point x) may create the appearance of a spatially based force -dV(x)/dx. This, however, has implications on the probability of the particle P(p,x). We try to investigate these ideas in this note.