The Quantum Spin Compass · AI Generator #GEN-4C56EF86

Physics · quantum-mechanics

Problem

A quantum engineer designs a novel compass using electron spin states. The device consists of two electrons trapped in separate quantum dots, each with spin-1/2. The electrons are initially prepared in an entangled singlet state ∣ψ⟩=12(∣↑↓⟩−∣↓↑⟩)|\psi\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle). When the compass is oriented at angle θ\theta relative to magnetic north, a uniform magnetic field B⃗=B0(sin⁡θx^+cos⁡θz^)\vec{B} = B_0(\sin\theta\hat{x} + \cos\theta\hat{z}) is applied to both electrons, where B0=0.1  TB_0 = 0.1\;\text{T}. The Hamiltonian for each electron in this field is H^=−μ⃗⋅B⃗\hat{H} = -\vec{\mu} \cdot \vec{B}, where μ⃗=−gsμBS⃗\vec{\mu} = -g_s\mu_B\vec{S} is the magnetic moment, gs=2g_s = 2, μB=9.274×10−24  J/T\mu_B = 9.274 \times 10^{-24}\;\text{J/T}, and S⃗\vec{S} is the spin operator. The compass reading is determined by measuring the probability that both electrons have spin up along the magnetic field direction. After the magnetic field is applied for time t=1.0×10−10  st = 1.0 \times 10^{-10}\;\text{s}, find the probability that both electrons are measured to have spin up along the field direction when θ=30°\theta = 30°. Take ℏ=1.055×10−34  J⋅s\hbar = 1.055 \times 10^{-34}\;\text{J·s}.

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