A positron (charge

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

A positron (charge q=+eq=+e, mass me=9.109×10−31 kgm_e=9.109\times10^{-31}\,\mathrm{kg}) is studied in two separate experiments. Ignore spin and treat the motion as non-relativistic throughout. ((a)) In a static, uniform field B⃗=Bz^\vec B=B\hat z with B=2.5 TB=2.5\,\mathrm{T}, described equally by the Landau gauge A⃗L=(0,Bx,0)\vec A_L=(0,Bx,0) and the symmetric gauge A⃗S=12(−By,Bx,0)\vec A_S=\tfrac12(-By,Bx,0), find the gauge function Λ(x,y)\Lambda(x,y) with A⃗S=A⃗L+∇Λ\vec A_S=\vec A_L+\nabla\Lambda and Λ(0,0)=0\Lambda(0,0)=0, write down the phase relating the state in the two gauges, and evaluate that phase angle at (x0,y0)=(15 nm,−6 nm)(x_0,y_0)=(15\,\mathrm{nm},-6\,\mathrm{nm}). ((b)) Still in this field, use only the commutator [π^x,π^y]=iℏqB[\hat\pi_x,\hat\pi_y]=i\hbar qB to build ladder operators for the motion transverse to B⃗\vec B and obtain the Landau spectrum. Give ℏωc\hbar\omega_c in meV, ℓB\ell_B in nm, and the energy E3E_3 of the n=3n=3 level, in meV. ((c)) In the second experiment the same positron passes through an interferometer whose two arms encircle a long, thin, shielded solenoid carrying flux Φ=0.6 Φ0\Phi=0.6\,\Phi_0, outside which the field is exactly zero, with Φ0=h/e=4.136×10−15 Wb\Phi_0=h/e=4.136\times10^{-15}\,\mathrm{Wb}. Derive δ\delta in terms of Φ\Phi and Φ0\Phi_0 and give its value modulo 2π2\pi in rad and the fringe shift as a fraction of one fringe.

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