2 - WIEN 2k

Dirac equation: H D and Ψ are 4-dimensionalFor a free particle (i.e. V = 0):Solution in the slowparticle limit (p=0)Non-relativistic limitdecouples Ψ 1from Ψ 2and Ψ 3from Ψ 4⎛ ε − mec⎜⎜ 0⎜ˆ⎜− pz⎝−zc m e2( pˆ+ ipˆ)2⎛φ⎞⎜↑⎟⎜ 0 ⎟, ⎜ ⎟⎜ 0 ⎟⎝ 0 ⎠y−0ε − m c( pˆ− ipˆ)zpˆParticles: up & downze⎛ 0 ⎞⎜↓2 ⎜φm ec , ⎜ 0⎜⎟ ⎟⎟⎟⎟ ⎝ 0 ⎠2y−− pˆ( pˆ+ ipˆ)zε + m c0−zey2c m e2−⎛ 0 ⎞⎜ ⎟⎜ 0 ⎟, ⎜ ↑χ ⎟⎜ ⎟⎝ 0 ⎠( pˆ− ipˆ)xpˆ0zε + m c−e2y⎞⎛Ψ⎟⎜⎟⎜Ψ⎟⎜⎟Ψ⎜⎠⎝Ψ1234⎛ 0 ⎞⎜2 ⎜ 0m ec , ⎜ 0⎜⎟ ⎟⎟⎟⎟ ↓⎝ χ ⎠⎞⎟⎟⎟ = 0⎟⎠Antiparticles: up & downFor a spherical potential V(r):⎛Φ ⎞Ψ = ⎜ ⎟⎝ χ ⎠=⎛⎜⎝− igfnκnκ( r)Υκσ( ) ⎟ ⎞r Υκσ⎠g nκ and f nκ are Radial functionsY κσ are angular-spin functionsj = l +κ = −sjs2( +1 2)s = +1,−1