An electron ( , ) moves in a uniform field ,

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 (Qe=−e=−1.602×10−19 CQ_{\rm e}=-\mathrm{e}=-1.602\times10^{-19}\,\mathrm{C}, me=9.109×10−31 kgm_{\rm e}=9.109\times10^{-31}\,\mathrm{kg}) moves in a uniform field B=Bz^\mathbf{B}=B\hat{\mathbf z}, B=0.30 TB=0.30\,\mathrm{T}, described in the gauge A=(0,Bx,0)\mathbf{A}=(0,Bx,0). Because yy is cyclic, pyp_y is conserved and labels the orbit, and the motion in xx alone obeys Hx=px2/2me+Veff(x)H_x=p_x^2/2m_{\rm e}+V_{\rm eff}(x) with Veff(x)=(py−QeBx)2/2meV_{\rm eff}(x)=(p_y-Q_{\rm e}Bx)^2/2m_{\rm e}. ((a)) For the label py=1.00×10−25 kg m s−1p_y=1.00\times10^{-25}\,\mathrm{kg\,m\,s^{-1}}, find the fixed point x0x_0 of the (x,px)(x,p_x) phase portrait and state whether it is a centre or a saddle, giving the reason from the sign of Veff′′V_{\rm eff}''. ((b)) A gauge function Λ(x,y)=cy\Lambda(x,y)=cy with c=0.20 T mc=0.20\,\mathrm{T\,m} is added, A′=A+∇Λ\mathbf{A}'=\mathbf{A}+\nabla\Lambda. Give the new label py′p_y' for the same physical orbit, and state whether the electron's velocity and the value of HH change.

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