Relativistic Corrections to the 1D Nonrelativistic Case for the Hulthen Potential
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint |
| Terbitan: |
, 2019
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3471844 |
Daftar Isi:
- In (1) an exact solution to the Klein Gordon equation with a scalar Hulthen potential V(x)= S exp(-ax) / (1-qexp(-a)) is obtained using the Nikiforov-Uvarov method. From the results, one may see that energy E goes as m + B(a,q)m + C(a,q) + O(1/m) etc. Often the second order term of E is of the order of 1/m, but such is not the case here. Nevertheless, if B(a,q) is small, one may still obtain the usual nonrelativistic relations. In this note, we examine the ground state for which a solution to the Klein Gordon equation may be found by inspection. We then consider the equations needed to compute various orders of the full energy solution in order to find nonrelativistic results. We find the B(a,q)m term is obtained from: 2m V(x) W(x) = -sqrt(e*e+2e*m) d/dx W(x). This is not the Schrodinger equation, but we show that the kinetic energy -1/2m d/dx d/dx W(x) associated with the wavefunction W(x) of this equation yields the nonrelativistic result, plus a term which can be dropped. For levels above the ground state, we show that 2m V(x) W(x) = -sqrt(e*e+2e*m) d/dx W(x) may be solved by the Nikiforov-Uvarov method. This seems to suggest that in the nonrelativistic case, it may be easier to solve an equation slightly different from the Schrodinger equation, but compatible with the relativistic equation. We also examine a second correction to the energy.