A Modern Approach to Quantum Mechanics · Problem 10.3

Time-Independent Perturbations — John S. Townsend

Physics — Quantum Mechanics · pionic-hydrogen-time-evolution

Problem

A negatively charged pion (a spin-0 particle) is bound to a proton forming a pionic hydrogen atom. At time t=0t = 0 the system is in the state ∣ψ⟩=12∣1,0,0⟩+12∣2,1,1⟩+12∣2,1,0⟩|\psi\rangle = \frac{1}{2}|1, 0, 0\rangle + \frac{1}{\sqrt{2}}|2, 1, 1\rangle + \frac{1}{2}|2, 1, 0\rangle In an external magnetic field in the zz direction, the Hamiltonian is given by H^=p^22μ−e2∣r^∣+ω0L^z\hat{H} = \frac{\hat{\mathbf{p}}^2}{2\mu} - \frac{e^2}{|\hat{\mathbf{r}}|} + \omega_0 \hat{L}_z where μ\mu is the reduced mass of the pion-proton system. (a) What is ∣ψ(t)⟩|\psi(t)\rangle, the state of the system at time tt? What is ⟨E⟩\langle E \rangle at time tt for this state? (b) What are ⟨Lx⟩\langle L_x \rangle and ⟨Lz⟩\langle L_z \rangle at time tt for this state?

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