Still Another Conservation Law — Question 9

Question 9

An ideal neutral-kaon mixing model rotates a strangeness state

Ignore decay and CP violation. Let $|K_1\rangle=(|K^0\rangle+|\bar K^0\rangle)/\sqrt2$ and $|K_2\rangle=(|K^0\rangle-|\bar K^0\rangle)/\sqrt2$ have energies $E_1,E_2$. Initially the state is $|K^0\rangle$. Define $\Delta E=E_2-E_1$.

1. Derive the probability to observe an antikaon at time t.

2. Find the first full-conversion time for $\Delta E=4.00\times10^{-6}$ eV using $\hbar=6.582\times10^{-16}$ eV s.

Question 9 - Solution

An ideal neutral-kaon mixing model rotates a strangeness state

(a) Method and calculation

$$|\psi(t)\rangle=\frac{e^{-iE_1t/\hbar}|K_1\rangle+e^{-iE_2t/\hbar}|K_2\rangle}{\sqrt2}.$$$$\boxed{P(K^0\to\bar K^0)=\frac14|e^{-iE_1t/\hbar}-e^{-iE_2t/\hbar}|^2=\sin^2\frac{\Delta Et}{2\hbar}}.$$

(b) Interpretation and further calculation

$$\Delta Et/(2\hbar)=\pi/2\Rightarrow\boxed{t_{\mathrm{full}}=\pi\hbar/|\Delta E|}.$$$$\boxed{t_{\mathrm{full}}=5.16949\times10^{-10}\,\mathrm s}.$$

Physical check

Real neutral kaons decay and exhibit CP violation. The supplied unitary two-state model isolates oscillation rather than predicting the full experimental time distribution.