Biomedical Engineering Reference
In-Depth Information
In addition, in order to satisfy Eq. (2.20), we must necessarily have
|
J
lr
| =|
J
rl
|
.Thisisrigorouslyequivalenttothedefinitionoftheclass
E
2D
given above.
In the cartesian basis, this means
ε
+
ε
−
=
J
2D
J
2D
=
(2.38)
with
ε
±
=
J
ll
±
(
J
lr
+
J
rl
)
/
2, and
=
i
(
J
lr
−
J
rl
)
/
2. The condition
for chirality
|
J
lr
| =|
J
rl
|
implies
ε
+
=
ε
−
and
=
0. This condition
for chirality also implies a stronger restriction.
Indeed, writing the non-diagonal coe
cients
J
lr
,
J
rl
in the polar
form
J
lr
=
ae
i
φ
and
J
rl
=
be
i
χ
(with
a
,
b
theamplitudesand
φ
,
χ
the
phases), wecan define the ratio
(
ε
+
−
ε
−
)
=
i
J
lr
−
J
rl
η
=
J
lr
+
J
rl
which thus becomes
η
=
i
1
−
b
2
a
2
b
a
1
+
a
2
−
2
sin(
φ
−
χ
).
(2.39)
b
2
b
2
a
2
1
+
The condition for chirality therefore implies:
Imag
=
0.
(2.40)
ε
+
−
ε
−
)
(
In the context of planar chirality, it is also useful to check
whether
E
2D
is closed with respect to the matrices addition and
multiplication.Sameasfor
E
o.a.
,itisobviousthatthisisnotthecase,
as by summing
αβ
γα
J
2D
=
(2.41)
and
αγ
βα
J
enant
=
(2.42)
2D
onegets
2
αβ
+
γ
γ
+
β
,
(2.43)
2
α
which does not belong to
E
2D
(i
n
deed, w
e
have
|
J
lr
|
=
|
J
rl
|
). Similarly, for the product of
J
2D
and
J
enant
2D
one obtain
α
which for the same reasons does not also
2
2
α
(
γ
+
β
)
α
(
γ
+
β
)
α
+
β
2
2
+
γ
belong to
E
2D
.
