PHY431 Lecture 6
1
3/24/2000
8.8.
Parity Conservation and Non-Conservation
The Parity operation, or spatial inversion, is an example of a discrete transformation. As shown before,
such operators, when applied twice, return the original situation:
P
2
=
1
. Such operators are therefore
both unitary and Hermitian, and thus observable. If
P
is a symmetry operation, then the Parity
P
(ei-
genvalue of
P
) of the system is a conserved quantity.
Parity, or mirror symmetry, has long been considered a symmetry transformation for any system.
Physical laws (describing interactions) were considered mirror symmetric as a matter of course. It
came as quite a nasty surprise, when first T.D. Lee and C.N. Yang in 1956 proposed that parity was
NOT a symmetry for the weak interactions, shortly thereafter confirmed by the experiment of C.S. Wu
et al. However, parity remains a good symmetry for the electromagnetic and the strong interaction.
The parity operator flips the direction (i.e. the sign) of any true (
polar
) vector:
,
,
,
,
,
...
θ
θ
→−
→−
→−
∇ →−∇
→−
P
P
P
P
P
r
r
p
p
E
E
!
!
"!
"!
(8.1)
However, “false” (known as
axial
or
pseudo-vectors)
do NOT change sign. Examples of pseudo-
vectors are vector-products of polar vectors:
L
=
r
×
p
; under
P
both
r
and
p
change sign, and thus the
sign of
L
remains unchanged! All angular momentum and spin vectors behave this way under
P
. Simi-
larly, scalar quantities normally do not change sign under
P
; however, it is easy to construct
pseudo-
scalars
, scalar quantities that flip sign under
P
; e.g. the combinations
J
⋅
p
or
∇⋅
B
.
The action of
P
on a wavefunction is thus:
(
)
2
1
( , )
(
, );
( , )
(
, )
( , )
t
t
t
t
t
ψ
ψ
ψ
ψ
ψ
−
=
−
=
−
=
⇒
=
=
P
r
r
P P
r
P
r
r
P
1
PP
(8.2)
Because
P
2
=
1
, the observable eigenvalues
P
of
P
must be real and either +1 (even) or
−
1 (odd): if
ψ
(
r
,
t
) is an eigenvector of
P
then:
(
)
(
)
(
)
2
2
( , )
( , )
( , )
( , )
( , )
1
1
real
t
P
t
P
t
P
t
t
P
P
ψ
ψ
ψ
ψ
ψ
=
=
=
=
⇒
=
⇒
= ±
P P
r
P
r
P
r
r
r
(8.3)
The wavefunctions
ψ
(
r
,
t
) and
ψ
(-
r
,
t
) in Eq.(8.2) do not have to be alike at all; only when Parity is a
symmetry
operation should there exist a close relationship between these functions. If
P
is a symmetry
operation, it follows that [
H
,
P
]=0. Then
ψ
(
r
,
t
) can be chosen as an eigenfunction of both
H
and
P
.
Then:
(
)
[
,
] 0
(
)
( )
( );
( )
(
)
(
)
( )
( )
(
)
E
E
E
E
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
=
−
=
=
⇒
−
=
−
=
=
=
−
H P
H
r
H
r
r
H P
r
H
r
r
PH
r
P
r
r
(8.4)
Thus, both
ψ
(
r
,
t
) and
ψ
(-
r
,
t
) satisfy the Schrödinger equation, with precisely the same eigenvalue.
That means that the two functions must be either describing two different systems that are degenerate
in energy, or describe the same system and be proportional to one another:
ψ
(-
r
,
t
)=
P
ψ
(
r
,
t
) =
P
ψ
(
r
,
t
),
with
P
=
±
1 as proven earlier.
Parity conservation is a multiplicative conservation law: e.g. the function
ψ
(
x
)=sin
2
x
has even parity:
2
2
(sin
)
(sin
)
(sin
)
( 1)
1,
and:
(sin
cos )
(sin
)
(cos )
1
x
x
x
x
x
x
x
=
= −
= +
=
= −
P
P
P
P
P
P
A general function does not necessarily have a specific parity (equivalent to saying that not every func-
tion is necessarily an eigenvalue of
P
):
P
(sin
x
+cos
x
) =
−
sin
x
+cos
x
≠
sin
x
+cos
x
.