“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
 InQM, 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
B0
ra
ra
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
B0
ra
ra
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!