Quantum Mechanics for Applied Physics

1. Quantum Mechanics for Applied Physics Lecture III Time dependent transition amplitude Electron in a magnetic field Aharonov Bohm effect and its applications

2. Two levels system | cb(t) | 2 = cos(ωt)2 where ω is the Rabi frequency

5. Classical Electrodynamics Potentials • micro/macroscopic electric field • micro/macroscopic magnetic field • charge/current density • Constitutive equations Charge conservation polarization magnetization speed of light

6. Energy & momentum • Point charge: , • Lorentz force • Energy balance in a space domain • EM energy density • Kinetic energy of charges • Poynting’s energy flux vector using identity

7. Electron in a Magnetic field Quantization P change to Popandr to rop Wave function Lev_Davidovich_Landau

8. is easy to check that To actually solve Schrödinger’s equation for an electron confined to a plane in a uniform perpendicular magnetic field, it is convenient to use the Landau gauge. H commutes with px, so H and px have a common set of eigenstates, taking ) .

9. Landau Levels Example of quantization measurement Effects of Landau levels are only observed when the mean thermal energy is smaller than the energy level separation, , meaning low temperatures and strong magnetic fields

10. Aharonov Bohm • The Aharonov-Bohm effect demonstrates that the electromagnetic potentials, rather than the electric and magnetic fields, are the fundamental quantities in quantum mechanics. • The necessary conditions to observe the Aharonov-Bohm effect, i.e. a shift of the diffraction pattern that varies periodically with B, are:  • There must be at least two interfering alternatives for the particle to arrive at the detector, and  • At least two of these interfering alternatives must enclose a shielded magnetic field and must be topologically distinct.  The animation shows the superposition of the waves for the case with (in blue) and without (in red) a magnetic field.

11. Magnetic Aharonov Bohm effect • Result of the requirement that quantum physics be invariant with respect to the gauge choice for the vector potential A. • This implies that a particle with electric charge q traveling along some path P in a region with zero magnetic field must acquire a phase which is in si units: • phase difference between any two paths with the same endpoints therefore determined by the magnetic fluxΦ through the area between the paths is given by:

12. Nature Physics 4, 205 (2008) Dong-In Chang, Gyong Luck Khym, Kicheon Kang, Yunchul Chung, Hu-Jong Lee, Minky Seo, Moty Heiblum, Diana Mahalu, Vladimir Umansky

13. Aharonov-Bohm Oscillations in Semiconductor Quantum Rings Now an international research team from the Nijmegen High Field Magnet Laboratory (the Netherlands), the Eindhoven University of Technology (the Netherlands), the University of Antwerp (Belgium), the University of Moldova (Moldova) and the the Institute of Microelectronics in Madrid (Spain) has succeeded to detect oscillatory currents carried by single electron states in a semiconductor quantum ring. These findings were published in the journal Physical Review Letters.