A particle of mass and charge moves in a uniform magnetic field along

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

Action and Symmetry, Ch Canonical transformations, generating functions and symplectic structure

Problem

A particle of mass m=1.0×10−26 kgm = 1.0\times10^{-26}\,\mathrm{kg} and charge Qe=1.6×10−19 CQ_{\rm e} = 1.6\times10^{-19}\,\mathrm{C} moves in a uniform magnetic field B=0.20 TB = 0.20\,\mathrm{T} along zz, its motion confined to the xx-yy plane (pz=0p_z = 0). In the Landau gauge of Chapter 13, A=(0,Bx,0)\mathbf{A} = (0, Bx, 0), the Hamiltonian is H=px22m+(py−QeBx)22m,H = \frac{p_x^2}{2m} + \frac{(p_y - Q_{\rm e}Bx)^2}{2m}, with yy cyclic, so pyp_y is a constant of the motion; take py=−1.6×10−21 kg m s−1p_y = -1.6\times10^{-21}\,\mathrm{kg\,m\,s^{-1}}. At t=0t=0 the particle is at x=0x=0 with px=0p_x=0. ((a)) Write HH in the form H=px2/2m+12mωc2(x−x0)2H = p_x^2/2m + \tfrac12 m\omega_{\rm c}^2(x-x_0)^2, and give the numerical values of the cyclotron frequency ωc=QeB/m\omega_{\rm c} = Q_{\rm e}B/m and the guiding-centre position x0x_0. ((b)) Let x′=x−x0x' = x - x_0. The generating function F1=12mωcx′2cot⁡QF_1 = \tfrac12 m\omega_{\rm c}x'^2\cot Q gives px=mωcx′cot⁡Qp_x = m\omega_{\rm c}x'\cot Q, P=12mωcx′2/sin⁡2QP = \tfrac12 m\omega_{\rm c}x'^2/\sin^2Q, hence x′=2P/(mωc)sin⁡Qx' = \sqrt{2P/(m\omega_{\rm c})}\sin Q, px=2mωcPcos⁡Qp_x = \sqrt{2m\omega_{\rm c}P}\cos Q, and K=H=ωcPK = H = \omega_{\rm c}P. Find the constant value of PP for this particle. ((c)) Find the smallest time t1>0t_1 > 0 at which the particle returns to the guiding centre, x(t1)=x0x(t_1) = x_0, and give pxp_x at that instant. ((d)) Confirm the value of PP found in part (b) by computing ∮px dx′\oint p_x\,\mathrm{d}x' directly, as the area of the ellipse traced by (x′,px)(x', p_x) over one cyclotron period, and dividing by 2π2\pi.

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