1 / 26

“Significance of Electromagnetic Potentials in the Quantum Theory”* The Aharanov-Bohm Effect

“Significance of Electromagnetic Potentials in the Quantum Theory”* The Aharanov-Bohm Effect. Chad A. Middleton MSC Physics Seminar February 17, 2011 *Y. Aharonov and D. Bohm, Phys. Rev. 115 , 485 (1959). D. J. Griffiths , Introduction to Quantum Mechanics , 2 nd ed., pps. 384-391.

infinity
Télécharger la présentation

“Significance of Electromagnetic Potentials in the Quantum Theory”* The Aharanov-Bohm Effect

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. “Significance of Electromagnetic Potentials in the Quantum Theory”*The Aharanov-Bohm Effect Chad A. Middleton MSC Physics Seminar February 17, 2011 *Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., pps. 384-391. D.J. Griffiths, Introduction to Electrodynamics, 3rd ed.

  2. http://www.nsf.gov/od/oia/activities/medals/2009/laureatephotos.jsphttp://www.nsf.gov/od/oia/activities/medals/2009/laureatephotos.jsp • Yakir Aharonov receives a 2009 National Medal of Science for his work in quantum physics which ranges from the Aharonov-Bohm effect to the notion of weak measurement.

  3. Outline… • Maxwell’s equations in terms of E & Bfields • Scalar and vector potentials in E&M • Maxwell’s equations in terms of the potentials • Schrödinger equation • Schrödinger equation with E&M • Aharonov-Bohm Effect • A simple example

  4. Maxwell’s equations in differential form (in vacuum) Gauss’ law for E-field Gauss’ law for B-field Faraday’s law Ampere’s law with Maxwell’s correction these plus the Lorentz force completely describe classical Electromagnetic Theory

  5. Taking the curl of the 3rd & 4th eqns (in free space when  = J = 0) yield.. The waveequations for the E-, B-fields with predicted wave speed Light = EM wave!

  6. Maxwell’s equations… Gauss’ law for E-field Gauss’ law for B-field Faraday’s law Ampere’s law with Maxwell’s correction Q: Can we write the Maxwell eqns in terms of potentials?

  7. E, B in terms of A, Φ… • Φ is the scalar potential • is the vector potential • Write the (2 remaining) Maxwell equations in terms of the potentials.

  8. Maxwell’s equations in terms of the scalar & vector potentials Gauss’ Law Ampere’s Law

  9. Gauge invariance of A, Φ.. Notice: E & B fields are invariant under the transformations: for any function • Show gauge invariance of E & B.

  10. Maxwell’s equations in terms of the scalar & vector potentials Gauss’ Law Ampere’s Law

  11. Coulomb gauge: Maxwell’s equations become.. Easy Hard • Gauss’ law is easy to solve for , Ampere’s law is hard to solve for

  12. Maxwell’s equations in terms of the scalar & vector potentials Gauss’ Law Ampere’s Law

  13. Lorentz gauge: Maxwell’s equations become.. • Lorentz gauge puts scalar and vector potentials on equal footing.

  14. Schrödinger equation for a particle of mass m where is the wave function with physical meaning given by: • How do we include E&M in QM?

  15. Schrödinger equation for a particle of mass m and charge q in an electromagnetic field • is the scalar potential • is the vector potential • In QM, the Hamiltonian is expressed in terms of and NOT .

  16. Gauge invariance of A, Φ.. Notice: E & B fields and the Schrödinger equation are invariant under the transformations: for any function • Since and differ only by a phase factor, they represent the same physical state.

  17. The Aharonov-Bohm Effect http://physicaplus.org.il/zope/home/en/1224031001/Tonomura_en In 1959, Y. Aharonov and D. Bohm showed that the vector potential affects the behavior of a charged particle, even in a region where the E & B fields are zero!

  18. A simple example: • Consider: • A long solenoid of radius a • A charged particle constrained to move in a circle of radius b, with a < b • Magnetic field of solenoid: Vector potential of solenoid? (in Coulomb gauge)

  19. A simple example: • Consider: • A long solenoid of radius a • A charged particle constrained to move in a circle of radius b,with a < b • Magnetic field of solenoid: Vector potential of solenoid? (in Coulomb gauge)

  20. Notice:The wave function for a bead on a wire is only a function of the azimuthal angle

  21. Notice:The wave function for a bead on a wire is only a function of the azimuthal angle The time-independent Schrödinger eqn takes the form..

  22. The time-independent Schrödinger equation yields a solution of the form.. where

  23. Notice:The wave function must satisfy the boundary condition this yields…

  24. Finally,solving for the energy… where • Notice: • positive (negative) values of n represent particle moving in the same (opposite) direction of I.

  25. Finally,solving for the energy… where • Notice: • positive (negative) values of n represent particle moving in the same (opposite) direction of I. • particle traveling in same direction as I has a lower energy than a particle traveling in the opposite direction.

  26. Finally,solving for the energy… where • Notice: • positive (negative) values of n represent particle moving in the same (opposite) direction of I. • particle traveling in same direction as I has a lower energy than a particle traveling in the opposite direction. • Allowed energies depend on the field inside the solenoid, even though the B-field at the location of the particle is zero!

More Related