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Lecture 2

Lecture 2. Magnetic Field: Classical Mechanics Aharonov -Bohm effect. Josep Planelles. The Hamiltonian of a particle in a magnetic field. Where does this Hamiltonian come from?. Classical Mechanics:. Conservative systems. Newton’s Law. Lagrange equation. Hamiltonian.

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Lecture 2

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  1. Lecture 2 Magnetic Field:Classical MechanicsAharonov-Bohm effect Josep Planelles

  2. The Hamiltonian of a particle in a magnetic field Where does this Hamiltonian come from?

  3. Classical Mechanics: Conservative systems Newton’s Law Lagrange equation Hamiltonian

  4. Velocity-dependent potentials Time-independent field No magnetic monopoles Lagrange function in this case? Guess: Is equivalent to ?

  5. Lagrangian kinematic momentum: canonical momentum: Hamiltonian

  6. Gauge ; We may select c : Coulomb Gauge :

  7. Hamiltonian (coulomb gauge) ;

  8. Axial magnetic field B & coulomb gauge ; gauge

  9. Confined electron pierced by a magnetic field Spherical confinement, axial symmetry Competition: quadratic vs. linear term

  10. Landau levels AB crossings Electron energy levels, nLM, R = 3 nm InAs NC Electron energy levels, nLM, R = 12 nm InAs NC Electron energy levels, nLM, GaAs/InAs/GaAs QDQW Electron in a spherical QD pierced by a magnetic field

  11. Electron in a QR pierced by a magnetic field AB crossings

  12. Aharonov-Bohm Effect Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485 • Classical mechanics: equations of motion can always be expressed in term of field alone. • Quantum mechanics: canonical formalism. Potentials cannot be eliminated. • An electron can be influenced by the potentials even if no fields act upon it.

  13. Semiconductor Quantum Rings Litographic rings GaAs/AlGaAs A.Fuhrer et al., Nature 413 (2001) 822; M. Bayer et al., Phys. Rev. Lett. 90 (2003) 186801. self-assembled rings InAs J.M.García et al., Appl. Phys. Lett. 71 (1997) 2014 T. Raz et al., Appl. Phys. Lett 82 (2003) 1706 B.C. Lee, C.P. Lee, Nanotech. 15 (2004) 848

  14. Toy-example : Quantum ring 1D

  15. Quantum ring 1D

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