Relativistic Corrections to the 1D Schrodinger Equation for the Infinite Potential Well

 In a previous note (1), we tried to develop an approach for dealing with relativistic corrections to the Schrodinger equation by considering the relativistic bound state wavefunction:

Wrel(x) = Integral dprel f(prel) exp(i prel x)  ((1))

with prel = p/sqrt(1-p*p/(m*m) where p is the nonrelativistic momentum. We then expanded ((1)) in powers of 1/m. Wrel(x) is a solution of a relativistic equation (not the Schrodinger equation) and so f(p) is also a solution of this relativistic equation, even if p is the nonrelativistic momentum. Thus, we defined  Wrel(x) approx equal Wnonrel(x) + W1(x) + Wb(x). Wb(x) is obtained from the expansion of ((1)) while W1(x) may be a functional correction. We then examined different relative equations (Klein-Gordon, Dirac with potentials) to find correction terms. In this note, we try to demonstrate that all 1/m powers of Wb(x) are 0 and:

Integral prel*prel f(prel) exp(i prel x) = RHS of Klein-Gordon equation = Integral p*p f(p) exp(i p x)

Thus, -d/dx d/dx for prel*prel carries over exactly to -d/dx d/dx for p*p without any expansions.

We apply the model of (1) to the case of the Klein-Gordon equation for a particle in an infinite well. We use exact relativistic results from (2) and expand these in powers of 1/m and compare with the results of (1).