Chemistry Reference
In-Depth Information
state, with no explicit time dependence, time inversion really means reversal of the
direction of motion, where all angular momenta will be changing sign, including the
“spinning” of the electrons. We shall denote this operator as
ϑ
. It has the following
properties:
ϑ(
=
ϑ
f
k
+
ϑ
|
f
k
+|
f
l
|
|
f
l
)
(2.24)
ϑc
cϑ
|
f
k
=¯
|
f
k
These properties are characteristic of an
anti-linear
operator. As a rationale for the
complex conjugation upon commutation with a multiplicative constant, we consider
a simple case-study of a stationary quantum state. The time-dependent Schrödinger
equation, describing the time evolution of a wavefunction,
Ψ
, defined by a Hamil-
tonian
H
, is given by
−
i
∂Ψ
∂t
=
H
Ψ
(2.25)
For a stationary state, the Hamiltonian is independent of time, and the wavefunction
is characterized by an eigenenergy,
E
; hence the right-hand side of the equation is
given by
H
Ψ
=
EΨ
. The solution for the stationary state then becomes
Ψ(t
0
)
exp
iE(t
−
t
0
)
Ψ(t)
=
−
(2.26)
Hence, the phase of a stationary state is “pulsating” at a frequency given by
E/
.
Now we demonstrate the anti-linear character, using Wigner's argument that a per-
fect looping in time would bring a system back to its original state.
2
Such a process
can be achieved by running backwards in time over a certain interval and then re-
turning to the original starting time. Let
T
t
represent a displacement in time toward
the future over an interval
t
, and
T
−
t
a displacement over the same interval but
toward the past. The consecutive action of
T
t
and
T
t
certainly describes a per-
−
fect loop in time, and thus we can write:
T
t
T
=
E
(2.27)
−
t
The reversal of the translation in time is the result of a reversal of the time variable.
We thus can apply the operator transformation under
ϑ
, in line with the previous
results in Sect.
1.3
:
T
−
t
=
ϑ T
t
ϑ
−
1
(2.28)
The complete loop can thus be written as follows:
T
t
ϑT
t
ϑ
−
1
=
E
(2.29)
2
Adapted from [
3
, Chap. 26].
