Chemistry Reference
In-Depth Information
Note that the action of time reversal on the spin functions precisely corresponds to
the
C
2
(C
2
)
. This result may be generalized,
in the sense that time reversal can be represented as the product of complex conju-
gation, denoted as
K
, and a unitary operator acting on the components of a function
space, which we shall denote by the unitary matrix
operator and thus is represented by
D
K
.
When this operator is applied twice, it must return the same state, except possibly
for a phase factor, say exp
(iκ)
. Following Wigner, we now show that the two cases
ϑ
2
U
. We thus write
ϑ
= U
1 are in fact the only possibilities. Hence, the phase factor can be only either
+1 (time-even state) or
=±
−
1 (time-odd state) [
11
, Chap. 26]. Taking time reversal
twice, we have
ϑ
2
=U
K
U
K
=U× U=
exp
(iκ)
I
(7.54)
U × U
T
Since
U
is unitary, we have
= I
. Comparing this result with the previous
expression, it follows that
U=
exp
(iκ)
U
T
(7.55)
Taking the transpose of both matrices in this equation will not affect the phase factor:
U
exp
(iκ)
U
T
=
(7.56)
Combining Eqs. (
7.55
) and (
7.56
) yields
U=
exp
(
2
iκ)
U
(7.57)
which implies that exp
(iκ)
1.
We shall now examine the effects of these two kinds of time-reversal symmetry
on quantum systems under time-even Hamiltonians, i.e., in the absence of external
magnetic fields.
=±
ϑ
2
•
1. In the case of a positive sign, it is always possible to write states that
are time invariant. Consider a state that is described by a complex wavefunction,
|
Ψ
=+
, which is an eigenfunction of a time-even Hamiltonian. The time-reversed
function,
ϑ
|
Ψ
, will thus also be an eigenfunction with the same eigenenergy. The
two functions will either coincide or be linearly independent. The latter case leads
to a state that is at least twofold-degenerate. Both components of this degeneracy
may transform as the complex-conjugate irreps of point groups such as the cyclic
groups or the
T
h
group. When
ϑ
2
1, it is always possible to recombine these
two degenerate states into two linear combinations that are invariant under time
reversal. It is indeed sufficient to project the real and imaginary parts of these
functions:
=+
√
2
|
1
|
=
+
|
φ
Ψ
ϑ
Ψ
(7.58)
√
2
|
=
−
i
|
χ
Ψ
−
ϑ
|
Ψ
