Spatial Flux and the Bound State One Dimensional Dirac Equation with a Scalar Potential
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint |
| Terbitan: |
, 2019
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3363143 |
Daftar Isi:
- In a number of previous notes, it has been suggested that nonrelativistic mechanics makes use of a function W(x,t), called the wavefunction, which behaves under transformations x-> x+a, t->t+b (a, b infinitesimal) like exp(ipx-iEt), but with p replaced with <p conditional > and E with <E conditional>, where <ip conditional > = i [Sum over p p fp(t) exp(ipx)]/W(x,t) = -i d/dx W(x,t). The conditional average may be extended to any function of p. (A similar expression holds for functions of E, but with exp(-iEt)). For the relativistic case, E*E=p*p + mo*mo (c=1). If one extends the idea of conditional averages, one may obtain the Klein Gordon equation. A question arises as to the role of the conditional average in the one dimensional Dirac equation with a scalar potential. In this note, we suggest the Dirac equation with a scalar potential may follow from ideas based on spatial flux [-i d/dx W(x,t) / W(x,t)]. First, one writes [E*E - (m+Vp(x))2 ] as a product of linear terms: [E+ m+Vp(x)] and [E-m-Vp(x)] where Vp(x) is the potential. Each of these is like a different average p. Thus, if -i d/dx W(x,t) / W(x,t) is like a flux equal to average p in the nonrelativistic case, one may suggest two flux functions, namely: -i d/dx V(x) / V(x) = [E-m-Vp(x)] and -i d/dx U(x) / U(x) =[E+ m+V(x)] Here E is a constant for bound states.Both of these expressions have nonrelativistic limits, i.e. E-m-Vp(x)= E nonrel - Vp(x) and E+m+Vp(x) = 2m. Considering the first, one has: E nonrel - Vp(x)= -i d/dx V(x) / V(x), but one wants d/dx d/dx W(x)/ W(x) for a kinetic type energy. One could set d/dx V= d/dx W, but this leaves the problem of the denominator. One may see that the 2m in the kinetic energy may come from E+m+Vp(x). Thus, the flux equations as they stand do not seem suitable. A possible remedy is to insert functions a(x) and a-1(x) in the following manner: -i d/dx V(x) / V(x) = a(x) [E-m-Vp(x)] and -i d/dx U(x) / U(x) = a-1(x)[E+ m+V(x)] ((1)) One may then attempt to solve for U(x) and V(x) and obtain the one dimensional Dirac equation with a scalar potential Vp(x). That is the objective of this note.