An electron (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

An electron (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 the first experiment the electron sits in a static, uniform field B⃗=Bz^\vec B=B\hat z with B=1.2 TB=1.2\,\mathrm{T}. This field is described equally well 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 written in the two gauges, and evaluate that phase angle at the point (x0,y0)=(10 nm,5 nm)(x_0,y_0)=(10\,\mathrm{nm},5\,\mathrm{nm}). ((b)) Still in the field of part (a), use only the commutator of the kinetic momenta, [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 energy levels. Give the cyclotron energy ℏωc\hbar\omega_c in meV, the magnetic length ℓB\ell_B in nm, and the energy E2E_2 of the n=2n=2 level, in meV. ((c)) In the second experiment the same electron is sent through an interferometer whose two arms encircle a long, thin, shielded solenoid carrying flux Φ=2.3 h/e\Phi=2.3\,h/e, with h/e=4.136×10−15 Wbh/e=4.136\times10^{-15}\,\mathrm{Wb}, outside which the field is exactly zero. Derive the Aharonov-Bohm phase difference δ\delta between the two arms in terms of Φ\Phi and the electron's flux quantum Φ0=h/q\Phi_0=h/q, and give its value modulo 2π2\pi in rad and the resulting fringe shift, with its sign, as a fraction of one fringe.

Related theory

Browse the problem library