Biomedical Engineering Reference
In-Depth Information
Erwin Schrödinger decided that the way ahead was to abandon the classical energy
expression and start again with the relativistically correct equation
ε
2
=
m
e
c
0
+
c
0
p
2
(13.38)
He then made the operator substitutions
j
h
2π
∂
∂
t
j
h
2π
∂
∂
x
ε
→
p
x
→−
etc.
to arrive at the Klein-Gordan equation
∂
2
∂
x
2
+
Ψ (
r
,
t
)
=
∂
2
∂
y
2
+
∂
2
∂
z
2
−
1
c
0
∂
2
∂
t
2
−
4π
2
m
e
c
0
h
2
0
(13.39)
In discussing relativistic matters, it is usual to write equations such as this in
four-vector
notation
. We define a four-vector with components
⎛
⎞
⎛
⎞
x
1
x
2
x
3
x
4
x
y
z
j
c
0
t
⎝
⎠
=
⎝
⎠
so that Equation (13.39) becomes
∂
2
∂
x
1
+
Ψ
(
x
1
,
x
2
,
x
3
,
x
4
)
∂
2
∂
x
2
+
∂
2
∂
x
3
+
∂
2
∂
x
4
−
4π
2
m
e
c
0
h
2
=
0
(13.40)
The Klein-Gordan equation is more satisfactory in that it has the desirable symmetry but
it turns out that it cannot describe electron spin. In the limit of low energy, it is equivalent
to the familiar Schrödinger equation.
Paul Dirac had the ingenious idea of working with a relativistic equation that was linear
in the vector components. He wrote
γ
1
Ψ (
x
1
,
x
2
,
x
3
,
x
4
)
∂
∂
x
1
+
∂
∂
x
2
+
∂
∂
x
3
+
∂
∂
x
4
−
2π
m
e
c
0
h
γ
2
γ
3
γ
4
=
0
(13.41)
where the multipliers γ
i
have to be determined. This equation is called the
Dirac equation.
Both the Schrödinger and the Klein-Gordan equation are second order, and it is usual to
manipulate the Dirac equation in order to give a corresponding second-order equation. This
can be done by operating on Equation (13.41) with the operator
∂
∂
x
1
+
∂
∂
x
2
+
∂
∂
x
3
+
∂
∂
x
4
+
2π
m
e
c
0
h
γ
1
γ
2
γ
3
γ
4
A little operator algebra shows that the multipliers have to satisfy
2f
i
=
j
γ
i
γ
j
+
γ
j
γ
i
=
=
0f
i
j
