588 10 Particle Physics – II
10.91 Let aK^0 beam be formed through a strong interaction likeπ−+p→K^0 +Λ.
Neither∣
∣K^0 〉nor∣
∣
∣K^0
〉
is an eigen state of|cp〉. However, linear combinations
can be formed.|Ks〉=1
√
2
(∣
∣K^0 〉+
∣
∣
∣K^0
〉)
cp|Ks〉=1
√
2
(∣∣
∣K^0
〉
+
∣
∣K^0 〉
)
=|Ks〉cp=+ 1|KL〉=
1
√
2
(∣
∣K^0
〉
−
∣
∣
∣K^0
〉)
cp|KL〉=1
√
2
(∣
∣
∣K^0
〉
−
∣
∣K^0
〉)
=−|KL〉
cp=− 1whileK^0 andK^0 are distinguished by their mode of production,Ksand
KL are distinguished by the mode of decay. Typical decays areKs →
π^0 π^0 ,π+π−,KL→π+π−π^0 ,πμν.
Att=0, the wave function of the system will have the formψ(0)=∣
∣K^0 〉=√^1
2
(|Ks〉+|KL〉)As time developsKsandKLamplitudes decay with their characteristic life-
times. The intensity ofKsorKLcomponents can be obtained by squaring
the appropriate coefficient inΨ(t). The amplitudes therefore contain a factor
e−iEt/which describes the time dependence of an energy eigen function in
quantum mechanics.
In the rest frame of theK^0 we can write the factore−iEt/ase−imc(^2) t/
,
where m is the mass. The complete wavefunction for the system can therefore
be written as
ψ(t)=
1
√
2
[
|Ks〉e
−t( 1
2 τs+imsc2
)
+|KL〉e−t(
21 τ
L+imLc^2
)]
=
1
√
2
e−imsc(^2) t/[
|Ks〉e−
2 tτ
s+|KL〉eiΔmc
(^2) t/]
whereΔm=ms−mLand we have neglected the factore−t/^2 τLwhich varies
slowly (τL∼= 70 τs). Reexpressing|K 1 〉and|K 2 〉in terms of
∣∣
K^0
〉
and