Introduction to Quantum Mechanics · Problem 4.36

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · oscillating_magnetic_field

Problem

An electron is at rest in an oscillating magnetic field B=B0cos⁡(ωt) k^\mathbf{B} = B_0 \cos(\omega t)\,\hat{k}, where B0B_0 and ω\omega are constants. (a) Construct the Hamiltonian matrix for this system. (b) The electron starts out (at t=0t = 0) in the spin-up state with respect to the xx axis (that is: χ(0)=χ+(x)\chi(0) = \chi^{(x)}_+). Determine χ(t)\chi(t) at any subsequent time. Beware: This is a time-dependent Hamiltonian, so you cannot get χ(t)\chi(t) in the usual way from stationary states. Fortunately, in this case you can solve the time-dependent Schrödinger equation (Equation 4.162) directly. (c) Find the probability of getting −ℏ/2-\hbar/2, if you measure SxS_x. Answer: sin⁡2(γB02ωsin⁡(ωt))\sin^2\left(\frac{\gamma B_0}{2\omega}\sin(\omega t)\right). (d) What is the minimum field (B0B_0) required to force a complete flip in SxS_x?

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