Franz Schwabl Quantum Mechanics 4th Edit · Problem 9.5

Addition of Angular Momenta

Physics — Quantum Mechanics · spin-precession-magnetic-field

Problem

Consider the precession of the spin of an electron in a homogeneous magnetic field (0,0,B)(0, 0, B). (a) Using the Hamiltonian H=(e0/mc)S⋅BH = (e_0/mc)\boldsymbol{S} \cdot \boldsymbol{B}, write down the equation of motion for the spin operator in the Heisenberg picture and give the solution for the initial condition S(t=0)=S(0)\boldsymbol{S}(t=0) = \boldsymbol{S}(0). The solution is Sz(t)=Sz(0),S_z(t) = S_z(0), Sx(t)=cos⁡ωt Sx(0)−sin⁡ωt Sy(0),S_x(t) = \cos\omega t\, S_x(0) - \sin\omega t\, S_y(0), Sy(t)=sin⁡ωt Sx(0)+cos⁡ωt Sy(0).S_y(t) = \sin\omega t\, S_x(0) + \cos\omega t\, S_y(0). (b) Given Ψ(0)=(ab)\Psi(0) = \binom{a}{b}, determine the state Ψ(t)\Psi(t) at the time tt. (c) What is the probability of obtaining the value ℏ/2\hbar/2 in a measurement of SzS_z at the time tt if at the time t=0t = 0 the spin was oriented in the xx-direction? (d) The same as in problem (c) for a measurement of SxS_x. (e) How can part (b) of this problem be derived from the Pauli equation?

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