the Aharanov-Bohm Effect - Colorado Mesa University
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“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.
http://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.
Outline…
Maxwell’s equations in terms of E & B fields
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
Maxwell’s equations in differential form
(in vacuum)
E
Gauss’ law for E-field
0
B 0
E
Gauss’ law for B-field
B
t
B 0 0
0
Faraday’s law
E
Ampere’s law with
Maxwell’s correction
t
0J
these plus
F q E v B
the Lorentz force completely describe
classical Electromagnetic Theory
Taking the curl of the 3rd & 4th eqns
(in free space when = J = 0) yield..
2
2
1
2 2 E 0
c t
2
2
1
2 2 B 0
c t
The wave equations for the
E-, B-fields with
predicted wave speed
c
1
0 0
3.0 10 m / s
8
Light = EM wave!
Maxwell’s equations…
E
Gauss’ law for E-field
0
B 0
E
Gauss’ law for B-field
B
t
B 0 0
0
Faraday’s law
E
Ampere’s law with
Maxwell’s correction
t
0J
Q: Can we write the Maxwell eqns in terms of potentials?
E, B in terms of A, Φ…
B A
E
A
t
Φ is the scalar potential
A is the vector potential
Write
the (2 remaining) Maxwell equations in terms of the
potentials.
Maxwell’s equations in terms of the
scalar & vector potentials
2
A
t
Gauss’ Law
0
2
2
0 0 2 A A 0 0
0 J
t
t
Ampere’s Law
Gauge invariance of A, Φ..
B A
E
A
t
Notice:
E & B fields
are invariant under the transformations:
t
for any function ( r , t )
A A A
Show gauge invariance of E & B.
Maxwell’s equations in terms of the
scalar & vector potentials
2
A
t
Gauss’ Law
0
2
2
0 0 2 A A 0 0
0 J
t
t
Ampere’s Law
Coulomb gauge: A 0
Maxwell’s equations become..
0
2
2
2
0J
0 0 2 A 0 0
t
t
Easy
Hard
Gauss’ law is easy to solve for ,
Ampere’s law is hard to solve for A
Maxwell’s equations in terms of the
scalar & vector potentials
2
A
t
Gauss’ Law
0
2
2
0 0 2 A A 0 0
0 J
t
t
Ampere’s Law
Lorentz gauge: A 0 0
t
0
Maxwell’s equations become..
2
2
0 0 2
t
0
2
2
0 0 2 A 0 J
t
Lorentz gauge puts scalar and vector potentials
on equal footing.
Schrödinger equation for a particle of mass m
2
2
V i
t
2 m
where is the wave function with physical meaning given by:
dP d x
*
3
How do we include E&M in QM?
Schrödinger equation for a particle of mass m
and charge q in an electromagnetic field
1
i qA
2 m
2
q i
t
is the scalar potential
A is the vector potential
InQM, the Hamiltonian is expressed in terms
of , A and NOT E , B .
Gauge invariance of A, Φ..
Notice:
E & B fields and the Schrödinger equation are
invariant under the transformations:
t
A A A
e
for any function ( r , t )
iq /
Since and differ only by a phase factor,
they represent the same physical state.
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!
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:
B B 0 kˆ
B0
ra
ra
Vector potential of solenoid?
(in Coulomb gauge)
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:
B B 0 kˆ
B0
ra
ra
Vector potential of solenoid?
(in Coulomb gauge)
A
B
2 r
eˆ ,
r a
Notice:
The wave function for a bead on a wire is only a
function of the azimuthal angle
( )
eˆ
as
1 d
b d
r b,
/2
Notice:
The wave function for a bead on a wire is only a
function of the azimuthal angle
( )
eˆ
as
r b,
/2
1 d
b d
The time-independent Schrödinger eqn takes the form..
2
2
2
2
1
d
q B d
q B
i
2
E
2
2
2 2
b d 4 b
2 m b d
The time-independent Schrödinger equation yields a
solution of the form..
( ) Ae
i
where
q B
2
b
2 mE
Notice:
The wave function must satisfy the boundary condition
( ) ( 2 )
e
i 2
1
this yields…
q B
2
b
2 mE n
where
n 0,1, 2, ...
Finally,
solving for the energy…
q B
where n 0,1,2,...
En
n
2
2 mb
2
2
2
Notice:
positive (negative) values
of n represent particle moving in the
same (opposite) direction of I.
Finally,
solving for the energy…
q B
where n 0,1,2,...
En
n
2
2 mb
2
2
2
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.
Finally,
solving for the energy…
q B
where n 0,1,2,...
En
n
2
2 mb
2
2
2
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!