An alpha particle ( , ) 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 alpha particle (Qe=2e=3.204×10−19 CQ_{\rm e}=2\mathrm{e}=3.204\times10^{-19}\,\mathrm{C}, m=6.645×10−27 kgm=6.645\times10^{-27}\,\mathrm{kg}) moves in a uniform field B=Bz^\mathbf{B}=B\hat{\mathbf z}, B=0.80 TB=0.80\,\mathrm{T}, together with a static rippled potential ϕ(x)=ϕ0cos⁡(qwx)\phi(x)=\phi_0\cos(q_{\rm w}x), ϕ0=150 V\phi_0=150\,\mathrm{V}, qw=2.00×103 rad m−1q_{\rm w}=2.00\times10^{3}\,\mathrm{rad\,m^{-1}}, in the gauge A=(0,Bx,0)\mathbf{A}=(0,Bx,0). ((a)) Give Veff(x)V_{\rm eff}(x) for the label py=0p_y=0, the condition on ϕ0\phi_0 for the crest at x=0x=0 to be a saddle rather than a minimum, and evaluate the threshold value of ϕ0\phi_0 numerically. Confirm the crest is a saddle here, and give Veff(0)V_{\rm eff}(0) in electronvolts. ((b)) The same field is written instead as A′=(0,Bx+c,0)\mathbf{A}'=(0,Bx+c,0), c=0.10 T mc=0.10\,\mathrm{T\,m} (also a uniform field Bz^B\hat{\mathbf z}). Find the new label py′p_y' of the orbit that had py=0p_y=0 in the first gauge, and state what happens to the shape and depth of VeffV_{\rm eff}.

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