Relativistic Corrections to the 1-D Schrodinger Equation
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint eJournal |
| Terbitan: |
, 2019
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3456957 |
Daftar Isi:
- CORRECTION: Eqaution ((5)) should be modified to: <prel*prel> = Integral dp p*p d/dp [p*p*p/(2m*m) exp(ipx) fp] + p*p*p*p/(m*m) Integral dp fp exp(ipx) This leads to -.5 p*p*p*p /(2m*m) Wo(x) + .25 p*p*p*p/ (2m*m) Wo + p*p*p*p/(m*m) Wo. This in turn affects ((11)) and ((15)) with p*p*p*p/(m*m) uo being replaced by the above. (Here p=hbar/i d/dx.) The two terms actually cancel, so further corrections need to be calculated. In addition, it seems e (the Schrodinger energy) in ((11)) and ((15)) should be replaced with e+de where de is a correction. In previous notes, it was suggested that -1/2m d/dx d/x W(x), where W(x) is the wavefunction, may be written as: 1/2m Sum over p [ p*p f(p) exp(ipx)] ((1)) and W(x)=Sum over p [f(p) exp(ipx)] ((2)). This result should hold for the relativistic case, thus one may use pRel = p/ sqrt(1- p*p/m*m) where p is the nonrelativistic momentum and c=1. One may then calculate expressions for ((1)) and ((2)) performing an expansion in 1/m to find W= Wa + Wb and <pRel*pRel> = p*p + a correction. This may be used in both the Klein Gordon and Dirac equations, leading to an Schrodinger like equation with extra terms. One may set Wa=Wa0+Wa1 where Wa0 is the Schrodinger solution, leaving a differential equation for Wa1. This DE depends on the relativistic equation used. Thus, one finally obtains: W=Wa0+Wa1+Wb and an altered Hamiltonian. One may use both of these to calculate energy shifts using: <Wa0+Wa1+Wb | H altered |Wa0+Wa1+Wb> - <Wa0| H | Wa0>. We consider the case of a Klein Gordon equation with both a scalar and vector potential, a Klein Gordon case with only a scalar potential and the Dirac equations. Given that some quantum mechanical bound particles move with speeds in the range of 107 m/s, it seems relativistic corrections may be important.