Elementary Particles
377
Now, if nature does not distinguish between the coordinate frame and the
inverted coordinate frame, i.e. space inversion symmetry exists, then ψ′(x, y, z)
also must satisfy the Schrödinger equation
( , , )
i
P x y z
t
∂ y
∂
= HPψ (x, y, z)
(10.35)
Since
P
2
= 1,
PHP = H
(10.36)
or
HP = PH
(10.37)
Thus, P commutes with the Hamiltonian. Therefore, parity is conserved
and states which are simultaneous eigenstates of H and P can be obtained
(See Eq. (3.42) and the dissuasion which follows it). This is valid provided
nature exhibits space-inversion symmetry (which has been shown to be
equivalent to having HP = PH).
In the fifties, two mesons called the τ-meson and the θ-meson were
discovered in cosmic rays. They have the same mass, around 498 MeV, the
same lifetime, around 1.2 × 10
–8
s, and the same production rates in nuclear
reactions of the type π
+
N → Λ + (τ
+
or θ
+
). However, they had different decay
modes: the τ-meson decayed into only two π-mesons (τ
+
→ 2π
+
+ π
–
) while the
θ-meson decayed into only two π-mesons (θ
+
→ π
+
+ π
0
). Now, the intrinsic
parity for a π-meson is –1 (as deduced from the strong interaction of the
π-mesons) so that p i = 1 for the two π−meson states. It was also shown by Dalitz
(1953) from the energy distribution of the pions, that p e = 1 for both two
π-meson final states. Thus, the decay products of the τ-meson decay are in a
negative parity state p i p e = –1, while the deacy products of the θ-meson decay
are in a positive parity state p i p e =1. If parity is conserved, an unusual situation
occurs, viz. that there are two particles with almost the same mass but opposite
party. The other possibility is that τ and θ are one and the same particle but the
weak interaction which is responsible for the decay (lifetimes indicate that the
interaction is weak), violates parity invariance, i.e. parity is not conserved in
weak interaction. Lee and Yang (1956) suggested this possibility after a critical
examination of processes involving weak interaction, and proposed an
experiment to test parity noncompensation in weak interaction.
Before describing the experiment, it is noted that space inversion is
equivalent to a reflection and a rotation, e.g. reflection in the xz plane
(change y → –y), and a rotation about the y-axis by 180° (changes x → – x and
z → –z). Since rotational invariance is a universal symmetry, it gives the result
that in addition to ψ′ (x, y, z) = Pψ (x, y, z) satisfying the Schrödinger equation,
the space-reflected wave function also satisfies the Schrödinger equation. e.g.
Rψ (x, y, z) where R denotes the operator which changes y → –y. This is known
as right-left symmetry. Consider. The nuclei of Co
60
whose spins are aligned
along the z-direction with the aid of a magnetic field. It was then found that the
377
Now, if nature does not distinguish between the coordinate frame and the
inverted coordinate frame, i.e. space inversion symmetry exists, then ψ′(x, y, z)
also must satisfy the Schrödinger equation
( , , )
i
P x y z
t
∂ y
∂
= HPψ (x, y, z)
(10.35)
Since
P
2
= 1,
PHP = H
(10.36)
or
HP = PH
(10.37)
Thus, P commutes with the Hamiltonian. Therefore, parity is conserved
and states which are simultaneous eigenstates of H and P can be obtained
(See Eq. (3.42) and the dissuasion which follows it). This is valid provided
nature exhibits space-inversion symmetry (which has been shown to be
equivalent to having HP = PH).
In the fifties, two mesons called the τ-meson and the θ-meson were
discovered in cosmic rays. They have the same mass, around 498 MeV, the
same lifetime, around 1.2 × 10
–8
s, and the same production rates in nuclear
reactions of the type π
+
N → Λ + (τ
+
or θ
+
). However, they had different decay
modes: the τ-meson decayed into only two π-mesons (τ
+
→ 2π
+
+ π
–
) while the
θ-meson decayed into only two π-mesons (θ
+
→ π
+
+ π
0
). Now, the intrinsic
parity for a π-meson is –1 (as deduced from the strong interaction of the
π-mesons) so that p i = 1 for the two π−meson states. It was also shown by Dalitz
(1953) from the energy distribution of the pions, that p e = 1 for both two
π-meson final states. Thus, the decay products of the τ-meson decay are in a
negative parity state p i p e = –1, while the deacy products of the θ-meson decay
are in a positive parity state p i p e =1. If parity is conserved, an unusual situation
occurs, viz. that there are two particles with almost the same mass but opposite
party. The other possibility is that τ and θ are one and the same particle but the
weak interaction which is responsible for the decay (lifetimes indicate that the
interaction is weak), violates parity invariance, i.e. parity is not conserved in
weak interaction. Lee and Yang (1956) suggested this possibility after a critical
examination of processes involving weak interaction, and proposed an
experiment to test parity noncompensation in weak interaction.
Before describing the experiment, it is noted that space inversion is
equivalent to a reflection and a rotation, e.g. reflection in the xz plane
(change y → –y), and a rotation about the y-axis by 180° (changes x → – x and
z → –z). Since rotational invariance is a universal symmetry, it gives the result
that in addition to ψ′ (x, y, z) = Pψ (x, y, z) satisfying the Schrödinger equation,
the space-reflected wave function also satisfies the Schrödinger equation. e.g.
Rψ (x, y, z) where R denotes the operator which changes y → –y. This is known
as right-left symmetry. Consider. The nuclei of Co
60
whose spins are aligned
along the z-direction with the aid of a magnetic field. It was then found that the