0.1 Klein-Gordon Equation 0.2 Dirac Equation

In higher order, the equivalence of DE to Pauli equation is approximate
because the energy eigenvalue appears in the expansion of the second term
in Eq.(10) so that is does not have the same form as the Pauli equation.
Nonetheless, we can obtain an approximate Pauli equation from the next
term in the expansion above as follows:
− 1
4m 2 (⃗σ · ⃗Π)(W + eφ)(⃗σ · ⃗Π) = − 1
4m 2 (⃗σ · ⃗Π) 2 (W + eφ) − 1
4m 2 (⃗σ · ⃗Π)[eφ, ⃗σ · ⃗Π].
It is correct to order v 4 to replace W + eφ in the first term on the right by
(⃗σ · ⃗Π) 2 according to Eq.(11). In the absence of a magnetic field ⃗ A = 0, the
result is the order v 4 relativistic kinetic energy correction −p 4 /8m 3 . Taking
the commutator in the second term and using the σ i σ j identity yields
second =
ie
4m (δ 2 ij + iɛ ijk σ k )Π i ∂ j φ
in which the first term is known as the Darwin term and the second is the
spin-orbit term which was added phonomenologically to the Pauli equation
in the olden days. To appreciate the physical significance of these terms, let’s
specialize to A ⃗ = 0, whence the Darwin term becomes
Darwin = − ie
4m 2 (∇2 φ − ⃗ E · ⃗∇).
This spin independent term represents a relativistic correction to the effective
potential which has obscure experimental consequences. For the pure
coulomb field, the first term is a delta function which shifts the energy of H-
atom s-states. The significance of the spin-orbit term appears if we specialize
to central potentials φ(r) where it becomes
SO = −
e 1 ∂φ
⃗σ · ⃗L.
4m 2 r ∂r
COMMENT
A more systematic and elegant way of approaching the NR approximation
is the Foldy-Wouthuysen (FW) transformation. The idea is to make a unitary
transformation of the states and operators, ψ = Uψ and O F W = UOU † with
UU † = 1, such that the new Hamiltonian is block diagonal, decoupling the
upper and lower components and giving separate two-component equations
for each. For a free particle, this is accomplished by
√
m + ω
U =
2ω + γ i p
√ i
,
2ω(ω + m)
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