in terms of it). In this pseudo-classical theoryp 1 ξ 1 +p 2 ξ 2 +p 3 ξ 3 is the function
generating a “supersymmetry”, Poisson commuting with the Hamiltonian, while
at the same time playing the role of a sort of “square root” of the Hamiltonian.
It provides a new sort of symmetry that can be thought of as a “square root”
of an infinitesimal time translation.
Quantization takes
p 1 ξ 1 +p 2 ξ 2 +p 3 ξ 3 →1
√
2
σ·Pand the Hamiltonian operator can now be written as an anticommutator or a
square
H=
1
2 m[
1
√
2
σ·P,1
√
2
σ·P]+=1
2 m(σ·P)^2 =1
2 m(P 12 +P 22 +P 32 )
(using the fact that theσjsatisfy the Clifford algebra relations for Cliff(3, 0 ,R)).
We will define the three dimensional Dirac operator as
∂/=σ 1∂
∂q 1+σ 2∂
∂q 2+σ 3∂
∂q 3=σ·∇It operates on two-component wavefunctions
ψ(q) =(
ψ 1 (q)
ψ 2 (q))
Using this Dirac operator (often called in this context the “Pauli operator”) we
can write a two-component version of the Schr ̈odinger equation (often called the
“Pauli equation” or “Pauli-Schr ̈odinger equation”)
i∂
∂t(
ψ 1 (q)
ψ 2 (q))
=−
1
2 m(
σ 1∂
∂q 1+σ 2∂
∂q 2+σ 3∂
∂q 3) 2 (
ψ 1 (q)
ψ 2 (q))
(34.3)
=−
1
2 m(
∂^2
∂q^21+
∂^2
∂q^22+
∂^2
∂q 32)(
ψ 1 (q)
ψ 2 (q))
This equation is two copies of the standard free particle Schr ̈odinger equation, so
physically corresponds to a quantum theory of two types of free particles of mass
m. It becomes much more non-trivial when a coupling to an electromagnetic
field is introduced, as will be seen in chapter 45.
The equation for the energy eigenfunctions of energy eigenvalueEwill be
1
2 m(σ·P)^2(
ψ 1 (q)
ψ 2 (q))
=E
(
ψ 1 (q)
ψ 2 (q))
In terms of the inverse Fourier transform
ψ 1 , 2 (q) =1
(2π)(^32)
∫
R^3eip·qψ ̃ 1 , 2 (p)d^3 p