Solutions of nonrelativistic Schrödinger equation from relativistic Klein–Gordon equation

Francesco R. Ruggeri

In a previous note, we argued that exp(i Action) for a nonrelativistic particle may be varied independently in T and X, where velocity = X/T. This variation indicates that energy is not kept constant. The solution exp(i Action)= exp(i .5m XX/TT) (for this case) leads to the nonrelativistic Schrodinger equation. In this note, we consider the free particle relativistic Lagragian (1): mo sqrt(1-vv) where v=X/T. We argue that the equation describing independent fluctuations of X and T is the Klein-Gordon equation.

Valentin V. Mdzinarishvili

Gian Fabrizio De Angelis, Maurizio Serva

P.-A. Gourdain

The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.

Frank V. Kowalski

Francesco R. Ruggeri

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.