PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Quantum Mechanics · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch Charged particles in electromagnetic fields
Problem
An electron (charge q=−e, mass me=9.109×10−31kg) moves in the plane across a uniform field B=Bz^, B=2.0T, when a uniform electric field E=Ex^, E=1.0×104Vm−1, is switched on. With ϕem=−Ex the transverse Hamiltonian becomes H^=2meπ^x2+π^y2−qEx^. Ignore spin and treat the motion as non-relativistic.
((a)) Write x^ in terms of the guiding-centre coordinate X^=x^+π^y/(qB) (with Y^=y^−π^x/(qB)) and π^y, and hence write H^ in terms of H^⊥, X^ and π^y.
((b)) Using the Heisenberg equation of motion, the guiding-centre commutator [X^,Y^]=−iℏ/(qB) and the fact that X^ and Y^ commute with π^x and π^y, find dX^/dt and dY^/dt. Show that the guiding centre drifts with a velocity that depends on none of q, me and ℏ, and give its direction.
((c)) Give the drift speed in ms−1, and the distance the guiding centre moves during one cyclotron period 2π/ωc, in nm.