Question

WKB expression for Dirac equation?

Answer 1

given the one dimensional Schroedinger equation

−ℏ22md2dx2Ψ(x)+V(x)Ψ(x)=EnΨ(x)−ℏ22md2dx2Ψ(x)+V(x)Ψ(x)=EnΨ(x)

the WKB method for the energies is

(n+1)2πℏ=∫baEn−V(x)−−−−−−−−−√dx(n+1)2πℏ=∫abEn−V(x)dx

with 'a' and 'b' being turning points

my question is what is the WKB aproximation for the Dirac equation in one or two dimension

(βmc2+∑k=13αkpkc)ψ(x,t)+V(x)Ψ(x,t)=iℏ∂ψ(x,t)∂t