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(?t)2 where ? is the Rabi
frequency
3
(No Transcript)
4
(No Transcript)
5
Classical Electrodynamics
Potentials
Charge conservation

• micro/macroscopic electric field
• micro/macroscopic magnetic field
• charge/current density
• Constitutive equations

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
• Poyntings energy flux vector

using identity
7
Electron in a Magnetic field
Quantization P change to Popand r to rop
Wave function
Lev_Davidovich_Landau
8
is easy to check that
To actually solve Schrödingers 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 flux F 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.