An electron (mass

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch The Hamiltonian, phase space and the variational principle in phase space

Problem

An electron (mass me=9.109×10−31 kgm_{\rm e} = 9.109\times10^{-31}\,\mathrm{kg}, charge Qe=−e=−1.602×10−19 CQ_{\rm e} = -\mathrm{e} = -1.602\times10^{-19}\,\mathrm{C}) moves at constant speed v=2.00×106 m s−1v = 2.00\times10^{6}\,\mathrm{m\,s^{-1}} on a circular orbit of radius rr centred on the origin, in a uniform magnetic field B=0.50 TB = 0.50\,\mathrm{T} along +z+z and with no electric field. At the point (r,0,0)(r,0,0) the electron's velocity is in the +y+y direction, since its negative charge makes it circle the opposite way to a proton. ((a)) Find rr. ((b)) Using the gauge A1=(0,Bx,0)\mathbf{A}_1 = (0, Bx, 0), find the canonical momentum pyp_y at (r,0,0)(r,0,0). ((c)) Using the gauge A2=12B(−y,x,0)\mathbf{A}_2 = \tfrac12B(-y,x,0), find pyp_y at the same point. ((d)) Evaluate the Hamiltonian H=∣p−QeA∣2/2meH = |\mathbf{p}-Q_{\rm e}\mathbf{A}|^2/2m_{\rm e} at that point in either gauge and confirm the two give the same value, and that this value equals the kinetic energy.

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