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Relativistic Corrections to the 1D Nonrelativistic Case for the Hulthen Potential

Francesco R. Ruggeri

Published Oct 3, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.3471843

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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(ee+2em) 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(ee+2em) 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.
The Klein Gordon equation for the harmonic oscillator may be written as (E-.5kx2)2=p2+mo2 ((1)) where mo is the rest mass and p the relativistic momentum. In the literature, it is argued that the resulting x4 term dominates for large x and prevents the Klein Gordon equation from being solved for bound states. As a result, it is replaced with an equation of the form: (E2-mo2) = (p-imwx)(p+imw) ((2)) where p=-ihbar d/dx in one dimension which may be solved exactly. Recently, the authors of (1) have used perturbation theory to calculate energy changes to ((1)) by using -.5V2 - e2 +2eV as a perturbative potential where V=.5kx2 and E=e+mo. They utilise the unperturbed wavefunction exp(-ax2) so large x4 values are damped. Their results are close to the exact calculations of ((2)) for the case of k=1 and m=1. In this note, we argue that for an oscillator there are classical turning points at which the quantum average kinetic energy -1/2m d/dx d/dx W should be zero and small nearby even if the average kinetic energy near x=0 corresponds to relativistic momenta. Thus, it is argued there are different spatial regions, each with its own quantum equation. The Schrodinger equation may be applied to regions near the classical turning point moving to infinite. In such a case, there is no issue with an x4 term. Near x=0, the full Klein Gordon equation ((1)) may be applied. There are questions of then matching the two sets of solutions. In this note, we attempt to investigate this scenario and compare results with those of (1) and (2). (1) and (2) suggest similar relativistic corrections to the nonrelativistic ground state oscillator energy of more than 20%. In this note, we suggest a smaller value.
In some recent articles in the literature (1),(2) the Schrodinger equation: p*p/2m W(x) + V(x) W(x) = E W(x) ((1)) is replaced with an equation using a relativistic momentum operator: [sqrt(p*p + m*m) - m] W(x) + V(x)W(x) = E W(x) ((2)) The first operator on the LHS of ((2)) is then expanded in powers of 1/(m*m). Next, p is replaced with hbar/i d/dx (or gradient in 3D) and perturbation theory used. In this note, we consider p*p as being a conditional average given by: p*p= [Integral dp f(p) exp(ipx) p*p ] / [Integral dp f(p) exp(ipx)] ((3)) The denominator is the wavefunction W(x). This result should hold for the nonrelativistic case. In the relativistic case, p is replaced everywhere with prel where prel= p/ sqrt(1-p*p/(m*m)) prel*prel = Integral dprel p*p (1+ p*p/(m*m)) f(prel) exp(iprel x) ((4)) And W(x) = Integral dprel f(prel) exp(iprel x) ((5)) To first order, we argue that Integral dprel f(prel) exp(iprel x) = Integral dp f(p) exp(i p x), but Integral p*p f(prel) dprel exp(i prel x) contributes a term . For higher orders, even Integral dprel f(prel) exp(i prel x) contributes. Thus, we try to argue that one does not only have a change in the prel*prel operator, but one must consider changes of prel within Integral dprel f(prel) exp(iprel x) itself. Thus, perturbation theory may not be enough in such cases.
In a previous note, the nonrelativistic bound state wavefunction W(x) was written as Wo(x)W1(x) with Wo(x) being the ground state wavefunction. W1(x) satisfies an eigenvalue equation, similar to the Schrodinger equation, but with the potential term replaced with a coupling term: -1/2m d/dx d/dx W1(x) - 1/m d/dx W1(x) [ d/dx Wo(x)/Wo(x) ] = (E-Eo) W1 (x) ((1)) Mathematically, ((1)) may be compared to the hypergeometric differential equation and solutions found for the Schrodinger if coefficients meet certain criteria. A similar approach applies to the Klein Gordon and Dirac equations In this note, we wish to examine possible physical significance of Wo(x)W1(x). In addition, we wish to obtain an equation, analogous to ((1)) for both the Klein Gordon and Dirac equations to compare the coupling terms.
We study the nonrelativistic Schrödinger equation for q-deformed Hulthen plus modified inversely quadratic Yukawa potential using the generalized parametric form of NIkiforov-Uvarov method. The energy eigenvalues and the corresponding normalized wave functions expressed in terms of hypergeometric function are obtained. We have also discussed four (4) special cases of this potential, i.e. Woods-Saxon, Hulthen, Morse and inversely quadratic potentials. Keywords: Nonrelativistic Schrödinger equation, q-deformed Hulthen potential, modified inversely quadratic Yukawa potential, parametric Nikiforov-Uvarov method.