Question

Show that the Klein-Gordan equation

h'c

and the Dirac equation

in the presence of electrogagnetic potentials A and ϕ are invariant under gauge transformations, provided the wave function is modified by a phase factor.

Answer 1

We know from Schrodinger equation H Phi = ih Partialdifferential Phi/Partialdifferential t and H^^= phi^2/2 m = -h^2 v^2/2m (for force particle) then -h^2/2m nabla^2 phi (r, t) = ih Partialdifferential Phi (r, t)/Partialdifferential t for relativistic particle H^^= E plusminus squareroot p^^2 c^2 + m^2_0 c^4 then plusminus [p^2 c^2 + m^2 c^4]^1/2 = ih Partialdifferential phi/Partialdifferential t (r, t) or (p^^2 c^2 + m^2 c^4) phi = -h^2 Partialdifferential^2 phi/Partialdifferential t^2 (r, t) p^^= ih (-h^2 c^2 nabla^2 + m^2 c^4) phi (r, t) = -h^2 Partialdifferential^2/Partialdifferential t^2 phi (r, t) k middot 4 e_r^n Dirac took a bolder approach by assuming the Hamiltonian H is linear in energy and momentum such that H = C alpha vector p + beta m c^2 then H^^u (r, t) = ih Partialdifferential phi (r, t)/Partialdifferential t (c alpha vector p + beta m c^2) phi (r, t)