588
10 Particle Physics – II
10.91 Let a K
0 beam be formed through a strong interaction like π
−
+ p → K
0
+Λ.
Neither
K
0
nor
K 0
is an eigen state of |cp. However, linear combinations
can be formed.
|K s =
1
√
2
K
0
+
K 0
cp |K s =
1
√
2
K 0
+
K
0
= |K s
cp = +1
|K L =
1
√
2
K
0
−
K 0
cp |K L =
1
√
2
K 0
−
K
0
= − |K L
cp = −1
while K
0 and K 0 are distinguished by their mode of production, K s and
K L are distinguished by the mode of decay. Typical decays are K s →
π
0
π
0
, π
+
π
−
, K L → π
+
π
−
π
0
, πμν.
At t = 0, the wave function of the system will have the form
ψ(0) =
K
0
=
1
√
2
(|K s + |K L )
As time develops K s and K L amplitudes decay with their characteristic lifetimes. The intensity of K s or K L components can be obtained by squaring
the appropriate coefficient in Ψ(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 the K
0 we can write the factor e
−iEt/ as e
−imc
2 t/ ,
where m is the mass. The complete wavefunction for the system can therefore
be written as
ψ(t) =
1
√
2
|K s e
−t
1
2τs +
ims c 2
+ |K L e
−t
1
2τ L
+
im L c 2
=
1
√
2
e
−im s c
2 t/
|K s e
−
t
2τs + |K L e
iΔmc
2 t/
where Δm = m s − m L and we have neglected the factor e
−t/2τL which varies
slowly (τ L ∼ = 70τ s ). Reexpressing |K 1 and |K 2 in terms of
K
0
and
K 0