| _version_ | 1866902061655785472 |
|---|---|
| author | Ruggeri, Francesco R. |
| author_facet | Ruggeri, Francesco R. |
| contents | <p><strong> </strong></p> <p dir="ltr"> In (1), the problem of one dimensional reflection-refraction is considered in terms of continuity of the photon electric field at an n1-n2 (index of refraction) point. From Maxwell’s equations, the electric field of a photon behaves as cos(-wt+kx), but exp(-iEt+ipx) is used to simplify the math. It is then argued that the electric field and its first derivative must be continuous at the junction point of n1-n2 which is taken to be x=0. These two equations allow one to solve for the coefficient of the reflected and refracted electric fields in terms of the incident electric field (whose weight we take to be 1). One may then combine the two equations to obtain a conservation of particle number equation: AA/c1 = BB/c1 + CC/c2 ((1)) where c1 is the speed in n1, c2 in n2 and AA, BB and CC are fluxes of the incident, reflected and refracted photons.</p> <p dir="ltr"> Here we argue that one may solve this problem strictly in terms of conservation. The conservation of particle number ((1)) is an obvious equation of conservation, but there are two unknowns if A=1 and the problem cannot be solved. This seems to suggest a second conservation equation, we argue. To find this conservation law, we note that there is a probability associated with free particles if one considers that a 2-body collision which is elastic has equal product probability for energy and momentum outcomes ei,ej and pxi, pxj if ei+ej=e1+e2 and pxi+pxj = px1+px2, where e1,e2 and px1,px2 are the initial values. This leads to a probability exp(-iEt+ipx) which we argue must also be conserved across the junction point. </p> <p dir="ltr">We link this probability to A exp(-iEt+ipx) where A*A = flux in the case of problems containing flux.We then show that one may solve for B and C from these two conservation equations without resort to the continuity of the first derivative equation or argument that one must have mathematically continuity of the electric field across the n1-n2 junction point. In other words, the problem may be seen entirely as a conservation one, but based on two conservation ideas.</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16989040 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | 1-D Reflection-Refraction in Terms of Conservation of Probability and Particle Number Ruggeri, Francesco R. <p><strong> </strong></p> <p dir="ltr"> In (1), the problem of one dimensional reflection-refraction is considered in terms of continuity of the photon electric field at an n1-n2 (index of refraction) point. From Maxwell’s equations, the electric field of a photon behaves as cos(-wt+kx), but exp(-iEt+ipx) is used to simplify the math. It is then argued that the electric field and its first derivative must be continuous at the junction point of n1-n2 which is taken to be x=0. These two equations allow one to solve for the coefficient of the reflected and refracted electric fields in terms of the incident electric field (whose weight we take to be 1). One may then combine the two equations to obtain a conservation of particle number equation: AA/c1 = BB/c1 + CC/c2 ((1)) where c1 is the speed in n1, c2 in n2 and AA, BB and CC are fluxes of the incident, reflected and refracted photons.</p> <p dir="ltr"> Here we argue that one may solve this problem strictly in terms of conservation. The conservation of particle number ((1)) is an obvious equation of conservation, but there are two unknowns if A=1 and the problem cannot be solved. This seems to suggest a second conservation equation, we argue. To find this conservation law, we note that there is a probability associated with free particles if one considers that a 2-body collision which is elastic has equal product probability for energy and momentum outcomes ei,ej and pxi, pxj if ei+ej=e1+e2 and pxi+pxj = px1+px2, where e1,e2 and px1,px2 are the initial values. This leads to a probability exp(-iEt+ipx) which we argue must also be conserved across the junction point. </p> <p dir="ltr">We link this probability to A exp(-iEt+ipx) where A*A = flux in the case of problems containing flux.We then show that one may solve for B and C from these two conservation equations without resort to the continuity of the first derivative equation or argument that one must have mathematically continuity of the electric field across the n1-n2 junction point. In other words, the problem may be seen entirely as a conservation one, but based on two conservation ideas.</p> <p> </p> |
| title | 1-D Reflection-Refraction in Terms of Conservation of Probability and Particle Number |
| url | https://doi.org/10.5281/zenodo.16989040 |