0.1 Klein-Gordon Equation 0.2 Dirac Equation

in which ω is the operator + √ ⃗p 2 + m 2 which commutes with ⃗p. It is easily
verified that UU † = 1, and a lengthy calculation shows that the FW
Hamiltonian reduces to
( )
ω 0
H F W =
.
0 −ω
Thus the upper components represent the positive energy solutions and the
lower components represent the negative energy solutions.
Although the momentum operator is not altered by the FW transformation,
since U commutes with ⃗p, the position operator is modified and
becomes a non-local operator in a way that is difficult to calculate in this
context. This is an indication of the fact that it is not possible to form a
state that is localized to a single point using positive energy solutions only.
At best, the paritcle can be localized to within a Compton wavelength m −1 .
It is possible to do a FW transformation for particle subject to potentials,
but it must be done sucessively for each higher order in v 2 . Omit it here.
0.6 H Atom
0.6.1 Hamiltonian
The relativistic treatment of the H-atom is a major success of the Dirac
theory. More generally, we treat a particle in a central field in the absence of
a magnetic field. In particular, we neglect the (weak) magnetic field due to
the nuclear magnetic moment. With the Dirac equation γ 0 (−i∂ 0 + eA 0 )ψ +
γ i ∂ i ψ + mψ = 0 as a starting point, we rewrite in the form of a Schrödinger
equation i∂ t ψ = Hψ with the Hamiltonian
H = γ 0 γ i p i − α r + γ0 m
( )
m −
α
⃗σ · ⃗p
=
r
⃗σ · ⃗p −m − α (12)
r
where α = e 2 /4π is the fine structure constant, value approximately 1/137.
−α/r can be replaced by any central potential −eφ(r) or V (r).
0.6.2 Conserved Quantities
As noted earlier, the total angular momentum ⃗ J is conserved. Therefore we
can have simultaneous eigenstates of J 2 , J z , and H. in addition, the parity
8