Chemistry Reference
In-Depth Information
which can be re-arranged to give
∂
∂
E
=
i
(1.19)
t
while taking the second derivative with respect to position
x
,wefind
2
p
2
∂
=−
2
x
2
∂
or
2
∂
p
2
2
=−
(1.20)
∂
x
2
Classically, the total energy
E
of a particle at
x
is just found by adding the
kinetic energy
T
p
2
=
/
2
m
and potential energy
V
at
x
:
p
2
2
m
+
E
=
V
(1.21)
Schrödinger assumed that if you multiply both sides of eq. (1.21) by the
wavefunction
, the equation still holds:
p
2
2
m
+
V
E
=
(1.22)
Then substituting eqs (1.19) and (1.20) into (1.22), Schrödinger postu-
lated that the wavefunction
obeys the second order partial differential
equation
2
2
m
∂
2
∂
∂
=−
i
+
V
(1.23)
t
∂
x
2
This is referred to as Schrödinger's time-dependent (wave) equation; the
'proof' of its validity comes from the wide range of experimental results
which it has predicted and interpreted.
Formany problems of interest, the potential
V
(
)
does not varywith time
and so we can separate out the position- and time-dependent parts of
x
:
e
−
i
Et
/
(
x
,
t
)
=
ψ(
x
)
(1.24)
Substituting eq. (1.24) in (1.23), and then dividing through by e
−
iEt
/
gives
2
d
2
−
ψ(
x
)
+
(
)ψ(
)
=
ψ(
)
V
x
x
E
x
(1.25a)
2
m
dx
2
