Question

The neutral kaons K⁰ and bar K⁰ are the charge conjugate of each other and are distinguished by their strangeness. However, they decay similarly and mixing can occur by a virtual process like K⁰ ⇔ π⁺ + π⁻ ⇔ bar K⁰. Starting with a pure beam of K⁰’s at t = 0 obtain the intensity of K⁰’s and bar K⁰’s at time t, in terms of the mean lifetimes τ_L(0.9 × 10⁻¹⁰ s) and τ_s(0.5 × 10⁻⁷s) for the components K_L and K_s, long lived and short lived respectively

Answer 1

Let a K⁰ beam be formed through a strong interaction like π⁻+p → K⁰+Λ. Neither |K⁰⟩ nor |K⁰⟩ is an eigen state of |cp⟩However, linear combinations can be formed.

while K⁰ and bar K⁰ are distinguished by their mode of production, K_s and K_L are distinguished by the mode of decay. Typical decays are K_s → π⁰π⁰, π⁺π⁻, K_L → π⁺π⁻π⁰, πμν. 

At t = 0, the wave function of the system will have the form

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/ℏ which describes the time dependence of an energy eigen function in quantum mechanics. 

In the rest frame of the K⁰ we can write the factor e^−iEt/ℏ as e^−imc2t/ℏ, where m is the mass. The complete wavefunction for the system can therefore be written as