Chemistry Reference
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that has non-trivial spinorial solutions only by canceling the associate determinant,
i.e.
, by forming the Equation
c
2
k
(∂
k
R)
σ
k
2
(∂
t
R)
2
=
ˆ
(10.16)
of which the minus sign of the squared root corresponds with the electronic conser-
vation charge, while the positive sign is specific to the relativistic treatment of the
positron motion. For proofing this, the specific relationship for the electronic charge
conservation (10.1) may be unfolded by adapting it to the present Bohmian spinorial
case by the chain equivalences
+∇
j
0
=
∂
t
ρ
k
∂
k
j
k
∂
t
(
R
2
)
=
+
k
∂
k
(
c
↔
0
ˆ
α
k
↔
0
)
=
2
R∂
t
R
+
⎡
⎣
⎤
⎦
i
¯
h
(S
+
s)
⎡
⎤
⎦
h
(S
+
s)
2
k
∂
k
R
∗
R
0
σ
k
e
i
¯
i
¯
c
⎣
=
2
R∂
t
R
+
e
−
h
(S
+
s)
e
i
¯
σ
k
0
ˆ
e
−
h
(S
+
s)
2
k
ˆ
c
σ
k
(
φ
2
ϕ
2
1
)
∂
k
R
2
=
2
R∂
t
R
+
1
+
2
Rc
k
ˆ
=
2
R∂
t
R
+
σ
k
(
∂
k
R
)
(10.17a)
The result
∂
t
R
=−
c
k
σ
k
(
∂
k
R
)
(10.17b)
indeed corresponds with the squaring root of (10.16) with the minus sign, certifying,
therefore, the validity of the present approach,
i.e.
, being in accordance with the step
(2) in the above bondonic algorithm.
Next, let us see what information is conveyed by the real part of Bohmian
decomposed spinors of Dirac Eq. (10.14); the system (10.18) is obtained
⎧
⎨
φc
k
(
∂
k
S
)
mc
2
+
+
−
σ
k
=
ˆ
ϕ
(
∂
t
S
w
)
0
(10.18)
ϕc
k
(
∂
k
S
)
⎩
mc
2
σ
k
−
ˆ
(
∂
t
S
+
+
w
)
φ
=
0
that, as was previously the case with the imaginary counterpart (10.15), has no trivial
spinors solutions only if the associate determinant vanishes, which gives the Equation
c
2
k
(∂
k
S) σ
k
2
=
∂
t
S
+
mc
2
w
2
+
(10.19)
