Chemistry Reference
In-Depth Information
The starting point resides in considering the de Broglie-Bohm electronic wave-
function (de Broglie and Vigier
1953
; Bohm and Vigier
1954
),
R
(
x
,
t
)exp
i
S
(
x
,
t
)
¯
BB
(
x
,
t
)
=
(11.1)
h
with the
R
-amplitude and
S
-phase factors given respectively as:
(
x
,
t
)
2
=
ρ
1
/
2
(
x
)
R
(
x
,
t
)
=
(11.2a)
S
(
x
,
t
)
=
px
−
Et
=
S
0
−
Et
(11.2b)
in terms of electronic density
ρ
, momentum
p
, total energy
E
, and space-time (
x,t
)
coordinates, without spin. In these conditions, since one perfumes the wavefunction
partial derivatives respecting space and time,
∂
2
R
∂x
2
2
exp
i
¯
∂S
∂x
h
S
∂
2
BB
∂x
2
2
i
¯
∂R
∂x
∂S
∂x
+
h
R
∂
2
S
i
¯
R
¯
=
+
−
(11.3a)
h
∂x
2
h
2
∂R
∂t
+
exp
i
¯
h
S
∂
BB
∂t
h
R
∂S
i
¯
=
(11.3b)
∂t
the conventional Schrödinger equation (Schrodinger
1926
)
h
2
2
m
∂
2
BB
∂x
2
h
∂
BB
∂t
=−
¯
i
+
V
BB
(11.4)
¯
takes the real and imaginary forms:
2
∂R
∂x
∂x
+
R
∂
2
S
∂R
∂t
=−
1
2
m
∂S
(11.5a)
∂x
2
∂S
∂x
2
h
2
2
m
∂
2
R
∂x
2
R
∂S
R
2
m
∂t
=−
¯
−
+
+
VR
(11.5b)
that can be further arranged as:
R
2
m
∂R
2
∂t
+
∂
∂x
∂S
∂x
=
0
(11.6a)
∂S
∂x
2
h
2
2
m
∂
2
R
∂x
2
∂S
∂t
−
1
R
1
2
m
¯
+
+
V
=
0
(11.6b)
Worth noting that the first Eq. (
11.6a
) recovers in 3D coordinates the charge current
(j) conservation law,
∂ρ
∂t
+∇
(
R
2
/m
)
−
∇
S
j
=
0, j
=
(11.7a)
