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“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.

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

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  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!

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