10.3 Solutions 589|Ks〉=1
√
2
(∣
∣K^0
〉
+
∣
∣
∣K^0
〉)
|KL〉=
1
√
2
(∣
∣K^0 〉−
∣
∣
∣K^0
〉)
ψ(t)=1
2
e−imsc(^2) t/[
e−t/^2 τs
(∣
∣K^0
〉
+
∣
∣
∣K^0
〉)
+eiΔmc(^2) t/(∣∣
K^0
〉
−
∣
∣
∣K^0
〉)]
=
1
2
e−imsc(^2) t/[∣∣
K^0
〉(
e−t/^2 τs+eiΔmc(^2) t/)
+
∣
∣
∣K^0
〉(
e−t/^2 τs−eiΔmc(^2) t/)]
The intensity of the component is obtained by taking the absolute square of
the coefficient of
∣
∣
∣K^0
〉
I
(∣∣
K^0
〉)
=
1
4
[
e−t/τs+ 1 + 2 e−t/^2 τscos(
Δmc^2 t/)]
Similarly,I∣
∣
∣(K^0
〉)
=
1
4
[
e−t/τs+ 1 − 2 e−t/^2 τscos(
Δmc^2 t/)]
10.92 Refering to Problem 10.91, theKLstate can be written as
|KL〉=1
√
2
(∣
∣K^0
〉
−
∣
∣
∣K^0
〉)
(1)
WhenKLenters the absorber, strong interactions would occur withK^0 (S=
+1) and∣
∣
∣K^0
〉
(S=−1) components of the beam of the originalK^0 beam
intensity, 50% has disappeared byKS-decay. The remainingKLcomponent
consists of 50%K^0. Upon traversing the material the existence ofK^0 with
S =−1 is revealed by the production of hyperons in a typical reaction,
K^0 +p→Λ+π+
WhileK^0 components can undergo elastic and charge-exchange scattering
only, theK^0 component can in addition participate in absorption processes
resulting in the hyperon production. The emergent beam from the slab will
then have theK^0 amplitude f∣
∣K^0 〉andK (^0) amplitude f
∣
∣
∣K^0
〉
with f <
f <1. The composition of the emergent beam from the slab is given by
modifying (1).1
√
2(
f∣
∣K^0
〉
−f∣
∣∣K 0
〉)
=
(
f+f)
2
√
2
(∣
∣K^0
〉
−
∣
∣
∣K^0
〉)
+
(
f−f)
2
√
2
(∣
∣K^0
〉
+
∣
∣
∣K^0
〉)
=
1
2
(
f+f)
|KL〉+
1
2
(
f−f