Principles of Quantum Mechanics · Problem 18.6

Time-Dependent Perturbation Theory — R. Shankar

Physics — Quantum Mechanics · selection-rules

Problem

Magnetic dipole transitions: The interaction of the electron spin with the magnetic field of the electromagnetic wave can stimulate transitions—this is the magnetic dipole contribution. Show that the transition rate for magnetic dipole transitions is R=ω3μ03πc3ℏ2∣⟨f∣μ^∣i⟩∣2R = \frac{\omega^3 \mu_0}{3\pi c^3 \hbar^2} |\langle f | \hat{\boldsymbol{\mu}} | i \rangle|^2 where μ^=−ge(e/2me)S\hat{\boldsymbol{\mu}} = -g_e (e/2m_e) \mathbf{S} is the magnetic dipole operator (with ge≈2g_e \approx 2). (a) Determine the selection rules for magnetic dipole transitions (in hydrogen, with nn, ll, mlm_l, msm_s as quantum numbers). Note: The wave function ψnlml\psi_{nlm_l} and the spin state ∣sms⟩|s m_s\rangle are independent. (b) Show that the rate for the hyperfine transition in hydrogen (∣1,0,0,↑↓⟩→∣1,0,0,↓↑⟩|1, 0, 0, \uparrow\downarrow\rangle \to |1, 0, 0, \downarrow\uparrow\rangle) is R=ω3μ0ge2e212πc3me2R = \frac{\omega^3 \mu_0 g_e^2 e^2}{12\pi c^3 m_e^2} [Hint: Use ⟨↑∣Sx∣↓⟩=ℏ/2\langle \uparrow | S_x | \downarrow \rangle = \hbar/2, and similarly for SyS_y.] (c) Calculate the lifetime of this state, in seconds. Compare it to the age of the universe (≈1.4×1010\approx 1.4 \times 10^{10} years) and to the lifetime of the 2p2p state (1.60×10−91.60 \times 10^{-9} s).

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