"Dirac equation 2"

psi1 = sqrt(E + m c^2) *
       (1, 0, c pz / (E + m c^2), (px + i py) c / (E + m c^2)) *
       exp(-i xi / hbar)

psi2 = sqrt(E + m c^2) *
       (0, 1, (px - i py) c / (E + m c^2), -pz c / (E + m c^2)) *
       exp(-i xi / hbar)

psi3 = sqrt(E + m c^2) *
       (pz c / (E + m c^2), (px + i py) c / (E + m c^2), 1, 0) *
       exp(i xi / hbar)

psi4 = sqrt(E + m c^2) *
       ((px - i py) c / (E + m c^2), -pz c / (E + m c^2), 0, 1) *
       exp(i xi / hbar)

xi = E t - px x - py y - pz z
E = sqrt(px^2 c^2 + py^2 c^2 + pz^2 c^2 + m^2 c^4)

gamma0 = ((1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,-1))
gamma1 = ((0,0,0,1),(0,0,1,0),(0,-1,0,0),(-1,0,0,0))
gamma2 = ((0,0,0,-i),(0,0,i,0),(0,i,0,0),(-i,0,0,0))
gamma3 = ((0,0,1,0),(0,0,0,-1),(-1,0,0,0),(0,1,0,0))

D(psi) = 1/c dot(gamma0,d(psi,t)) +
         dot(gamma1,d(psi,x)) +
         dot(gamma2,d(psi,y)) +
         dot(gamma3,d(psi,z))

"Verify wavefunctions"

check(i hbar D(psi1) == m c psi1)
check(i hbar D(psi2) == m c psi2)
check(i hbar D(psi3) == m c psi3)
check(i hbar D(psi4) == m c psi4)

"ok"

"Verify normalization"

check(dot(psi1,conj(psi1)) == 2 E)
check(dot(psi2,conj(psi2)) == 2 E)
check(dot(psi3,conj(psi3)) == 2 E)
check(dot(psi4,conj(psi4)) == 2 E)

"ok"