The spin-zero neutral kaon is a system with two basis states, the eigenstates of $\sigma_z$, representing a particle $K^0$ and its antiparticle $\bar{K}^0$ : The operator $\sigma_x=C P$ represents the combined parity $(P)$ and charge conjugation $(C)$, or particle-antiparticle, transformation and takes $\alpha=\left|K^0\right\rangle$ into $\beta=\left|\bar{K}^0\right\rangle$. The dynamics is governed by the Hamiltonian matrix
$$
H=M-i \frac{\Gamma}{2}
$$
where $M$ and $\Gamma$ are Hermitian $2 \times 2$ matrices, representing the mass-energy and decay properties of the system, respectively. ${ }^7$ The matrix $\Gamma$ is positive definite. A fundamental symmetry (under the combined $C P$ and time reversal transformations) requires that $\sigma_x M^*=M \sigma_x$ and $\sigma_x \Gamma^*=\Gamma \sigma_x$.
(a) Show that in the expansion of $H$ in terms of the Pauli matrices, the matrix $\sigma_z$ is absent. Derive the eigenvalues and eigenstates of $H$ in terms of the matrix elements of $M$ and $\Gamma$. Are the eigenstates orthogonal?
(b) Assuming, as is the case to good approximation, that the Hamiltonian also satisfies the $C P$ invariance conditions $\sigma_x M=M \sigma_x$ and $\sigma_x \Gamma=\Gamma \sigma_x$ show that $H$ is normal, and construct its eigenstates, $\left|K_1^0\right\rangle$ and $\left|K_2^0\right\rangle$. If the measured lifetimes for these two decaying states are $\tau_1=\hbar / \Gamma_1=0.9 \times 10^{-10} \mathrm{sec}$ and $\tau_2=\hbar / \Gamma_2=0.5 \times$ $10^{-7} \mathrm{sec}$, respectively, and if their mass difference is $m_2-m_1=3.5 \times 10^{-6} \mathrm{eV} / \mathrm{c}^2$, determine the numerical values of the matrix elements of $M$ and $\Gamma$ as far as possible.
(c) If the kaon is produced in the state $K^0$ at $t=0$, calculate the probability of finding it still to be a $K^0$ at a later time $t$. What is the probability that it will be found in the $\bar{K}^0$ state? Plot these probabilities, exhibiting particle-antiparticle oscillations, as a function of time.