Robert Mann - An Introduction to Particle Physics and the Standard Model · Problem 6.7

Accelerators — Robert Mann

Physics — Particle Physics · time-reversal-operator-kramers-degeneracy

Problem

Time reversal interchanges initial and final states, so that if T∣χ⟩=∣χ⟩t\mathbb{T}|\chi\rangle = |\chi\rangle^{\mathbf{t}} then T(⟨ϑ∣χ⟩)=t⟨ϑ∣χ⟩t=⟨χ∣ϑ⟩=(⟨ϑ∣χ⟩)∗\mathbb{T}(\langle\vartheta|\chi\rangle) =^{\mathbf{t}} \langle\vartheta|\chi\rangle^{\mathbf{t}} = \langle\chi|\vartheta\rangle = (\langle\vartheta|\chi\rangle)^* where the first equality is due to interchange of initial and final states and the second equality is the property of quantum-mechanical amplitudes. (a) Show that the above relation implies Tc=c∗T\mathbb{T}c = c^*\mathbb{T} where cc is any complex number. (b) Given that TJ⃗=−J⃗T\mathbb{T}\vec{J} = -\vec{J}\mathbb{T} where J⃗\vec{J} is the angular momentum operator, show that ∣χ1⟩≡−T∣12,−12⟩and∣χ2⟩≡T∣12,12⟩|\chi_1\rangle \equiv -\mathbb{T}\left|\frac{1}{2}, -\frac{1}{2}\right\rangle \quad \text{and} \quad |\chi_2\rangle \equiv \mathbb{T}\left|\frac{1}{2}, \frac{1}{2}\right\rangle form a spin-1/21/2 in the time-reversed system. (c) What is T2\mathbb{T}^2 for a state ∣ϕ⟩|\phi\rangle with an odd number of spin-1/21/2 particles? What is T2\mathbb{T}^2 for a state ∣φ⟩|\varphi\rangle with an even number of spin-1/21/2 particles? (d) Suppose the state ∣ϕ⟩|\phi\rangle in (c) is an eigenstate of the Hamiltonian. What is the degree of the degeneracy of this state? (i.e. how many distinct states are there at a given energy level?) This degeneracy is called Kramer's degeneracy. (e) Is Kramer's degeneracy preserved in an electric field? Is it preserved in a magnetic field?

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