Magnetic Field: Classical Mechanics Aharonov-Bohm effect 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

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

Lagrangian kinematic momentum: canonical momentum: Hamiltonian

Gauge ; We may select c : Coulomb Gauge :

Hamiltonian (coulomb gauge) ;

Axial magnetic field B & coulomb gauge ; gauge

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

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

Electron in a QR pierced by a magnetic field AB crossings

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. In their celebrated 1959 paper[1] Aharonov and Bohm pointed out that while the fundamental equations of motion in classical mechanics can always be expressed in term of field alone, in quantum mechanics the canonical formalism is necessary, and as a result, the potentials cannot be eliminated from the basic equations. They proposed several experiments and show that an electron can be influenced by the potentials even if no fields acts upon it. More precisely, in a field-free multiple-connected region of space, the physical properties of a system depends on the potentials through the gauge-invariant quantity H Adl, where A represents the potential vector.

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 Los primeros QRs fueron fabricados en los 80 por técnicas litográficas. Mesoscópicos. Dispersión. Primeros anillos nanoscópicos recientes: litográficos por Fuhrer y Bayer; auto-ordenados por Jorge García. Éstos crecen por una ligera variación en la síntesis de QDs auto-ordenados. 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

Toy-example : Quantum ring 1D

Quantum ring 1D