Introduction to Quantum Mechanics · Problem 11.29

Quantum Dynamics — David J. Griffiths

Physics — Quantum Mechanics · magnetic-resonance

Problem

Magnetic resonance. A spin-1/2 particle with gyromagnetic ratio γ\gamma, at rest in a static magnetic field B0k^B_0\hat{k}, precesses at the Larmor frequency ω0=γB0\omega_0 = \gamma B_0 (Example 4.3). Now we turn on a small transverse radiofrequency (rf) field, Brf[cos⁡(ωt)i^−sin⁡(ωt)j^]B_{\rm rf}\left[\cos(\omega t)\hat{i} - \sin(\omega t)\hat{j}\right], so that the total field is B=Brfcos⁡(ωt)i^−Brfsin⁡(ωt)j^+B0k^.(11.123)\mathbf{B} = B_{\rm rf}\cos(\omega t)\hat{i} - B_{\rm rf}\sin(\omega t)\hat{j} + B_0\hat{k}. \tag{11.123} (a) Construct the 2×22 \times 2 Hamiltonian matrix (Equation 4.158) for this system. (b) If χ(t)=(a(t)b(t))\chi(t) = \begin{pmatrix} a(t) \\ b(t) \end{pmatrix} is the spin state at time tt, show that a˙=i2(Ωeiωtb+ω0a);b˙=i2(Ωe−iωta−ω0b),(11.124)\dot{a} = \frac{i}{2}\left(\Omega e^{i\omega t}b + \omega_0 a\right); \quad \dot{b} = \frac{i}{2}\left(\Omega e^{-i\omega t}a - \omega_0 b\right), \tag{11.124} where Ω≡γBrf\Omega \equiv \gamma B_{\rm rf} is related to the strength of the rf field. (c) Check that the general solution for a(t)a(t) and b(t)b(t), in terms of their initial values a0a_0 and b0b_0, is a(t)={a0cos⁡(ω′t/2)+iω′[a0(ω0−ω)+b0Ω]sin⁡(ω′t/2)}eiωt/2a(t) = \left\{a_0\cos(\omega' t/2) + \frac{i}{\omega'}\left[a_0(\omega_0 - \omega) + b_0\Omega\right]\sin(\omega' t/2)\right\}e^{i\omega t/2} b(t)={b0cos⁡(ω′t/2)+iω′[b0(ω−ω0)+a0Ω]sin⁡(ω′t/2)}e−iωt/2b(t) = \left\{b_0\cos(\omega' t/2) + \frac{i}{\omega'}\left[b_0(\omega - \omega_0) + a_0\Omega\right]\sin(\omega' t/2)\right\}e^{-i\omega t/2} where ω′≡(ω−ω0)2+Ω2.(11.125)\omega' \equiv \sqrt{(\omega - \omega_0)^2 + \Omega^2}. \tag{11.125} (d) If the particle starts out with spin up (i.e. a0=1a_0 = 1, b0=0b_0 = 0), find the probability of a transition to spin down, as a function of time. *Answer:* P(t)={Ω2/[(ω−ω0)2+Ω2]}sin⁡2(ω′t/2)P(t) = \left\{\Omega^2/\left[(\omega-\omega_0)^2+\Omega^2\right]\right\}\sin^2(\omega' t/2). (e) Sketch the resonance curve, P(ω)=Ω2(ω−ω0)2+Ω2,(11.126)P(\omega) = \frac{\Omega^2}{(\omega - \omega_0)^2 + \Omega^2}, \tag{11.126} as a function of the driving frequency ω\omega (for fixed ω0\omega_0 and Ω\Omega). Note that the maximum occurs at ω=ω0\omega = \omega_0. Find the "full width at half maximum," Δω\Delta\omega. (f) Since ω0=γB0\omega_0 = \gamma B_0, we can use the experimentally observed resonance to determine the magnetic dipole moment of the particle. In a nuclear magnetic resonance (nmr) experiment the gg-factor of the proton is to be measured, using a static field of 10,000 gauss and an rf field of amplitude 0.01 gauss. What will the resonant frequency be? (See Section 7.5 for the magnetic moment of the proton.) Find the width of the resonance curve. (Give your answers in Hz.)

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