The spin-zero neutral kaon is a system with two basis...

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.

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.

Key Concepts

Non-Hermitian Hamiltonians and Decay

When a quantum system involves unstable (decaying) states, its effective Hamiltonian becomes non-Hermitian. This non-Hermiticity arises from the inclusion of complex terms—typically seen as a combination of a Hermitian mass matrix and an anti-Hermitian decay matrix—which account for loss of probability (decay) in the system over time. This formalism is essential for describing systems with finite lifetimes.

Two-State Quantum Systems

This concept involves describing a quantum system with only two possible states, which can be represented as basis vectors in a two?dimensional Hilbert space. It is a fundamental framework for capturing phenomena such as particle-antiparticle mixing, qubit behavior in quantum computing, and similar processes where only two configurations are relevant.

Pauli Matrix Expansion

Any 2×2 matrix can be expressed in terms of the identity matrix and the three Pauli matrices (?x, ?y, ?z). This expansion is a powerful tool because it translates operator problems into an algebraic framework using well-understood basis elements, thereby simplifying the analysis of quantum systems and unveiling symmetries or specific features (e.g., the absence of a ?z term) in the physical Hamiltonian.

CP Symmetry in Particle Physics

CP symmetry combines charge conjugation (C), which transforms particles into their corresponding antiparticles, and parity (P), which inverts spatial coordinates. In particle physics, enforcing CP symmetry on the Hamiltonian (or its components) imposes constraints on the system’s dynamics, such as specific transformation properties of the mass-energy and decay matrices. This has major implications for mixing and decay processes in systems that involve particles and antiparticles.

Matrix Diagonalization and Eigenstates

Diagonalizing a Hamiltonian, whether Hermitian or non-Hermitian, involves finding its eigenvalues and corresponding eigenstates. This process lays the groundwork for understanding the energy spectrum, decay rates, and the dynamics of state evolution. For non-Hermitian operators, special care is needed in terms of the orthogonality and completeness of the eigenstates, as they may not be orthogonal under the usual inner product.

Normal Operators

A normal operator is one that commutes with its Hermitian conjugate. When a Hamiltonian satisfies this property, it guarantees that it can be diagonalized by a unitary transformation and that its eigenstates form an orthonormal basis. This is critical in quantum mechanics for ensuring predictable time evolution and consistency in probability interpretations.

Time Evolution and Quantum Oscillations

The time-dependent behavior of quantum states, particularly in two-state systems, leads to phenomena such as oscillations between different states. Using the eigenstates and eigenvalues of the Hamiltonian, one can compute the time evolution of an initially prepared state and calculate transition probabilities. This is particularly important in understanding the interference effects and oscillatory behavior observed in particle-antiparticle mixing.

Decay Widths and Particle Lifetimes

Decay width is a parameter that quantifies the rate at which an unstable quantum state decays, and it is directly related to the lifetime of the state (? = ?/?). By linking measured lifetimes to the decay matrix elements, one can extract important information about the intrinsic properties of the system, which is essential in matching theoretical predictions with experimental observations.

Transcript

00:01 For this problem on the topic of particle physics, we have a positive kion, which is a bound state of an up quark and an anti -down quark, decaying into an anti -meon and a muon neutrino, which is a process that is mediated by a w plus boson.

00:16 Now we want to use relativistic methods and solve for various properties of the kion, muon, and neutrino.

00:23 So firstly, for part a, by energy conservation, we have the mass of the kion mk times c -square.

00:31 Is equal to gamma times the mass of the mu on mmu times c squared plus the energy of the neutrino e new.

00:52 For part b by momentum conservation, we know the initial momentum is zero.

00:58 The neutrino is treated as massless, so we have the momentum of the neutrino equal to its energy divided by c and momentum conservation gives gamma m mu times v equal to e nu over c.

01:25 For part c we want the speed of the muon.

01:38 We'll combine the engine momentum equation from a and b, which gives us m k c squared equal to gamma m mu c squared plus gamma gamma m mue times v times c next we'll substitute for sigma and then for gamma and finally solve for v so sigma is equal to gamma into one plus v over c which is one over the square root of one minus v squared over c squared into 1 plus v over c.

02:26 And so by rearranging we get v to be c times sigma squared minus 1 over sigma squared plus 1.

02:45 For part d, sigma is equal to the mass of the k on over the mass of the muon, which would be given values is 493 .0 .7 mega -electro.

03:01 Volts by c squared divided by 105 .7 mega electron volts by c squared, which is a ratio of 4 .671.

03:25 For part e, we want e mu, and we'll use the results of c and d to find v, and we get v is equal to c into 4 .671 squared minus 1 divided by 4 .671 squared plus 1...

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