1 Gauge Transformation in Quantum Mechanics

Gauge transformation in quantum mechanics; Aharonov-Bohm effct

Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: April 29, 2015)

______David Joseph Bohm (20 December 1917 – 27 October 1992) was an American-born British quantum physicist who made contributions in the fields of theoretical physics, philosophy and neuropsychology, and to the Manhattan Project.

http://en.wikipedia.org/wiki/David_Bohm ______Yakir Aharonov (born 1932 in Haifa, Israel) is an Israeli physicist specializing in Quantum Physics. He is a Professor of Theoretical Physics and the James J. Farley Professor of Natural Philosophy at Chapman University in California. He is also a distinguished professor in Perimeter Institute. He also serves as a professor emeritus at Tel Aviv University in Israel. He is president of the Iyar, The Israeli Institute for Advanced Research. His research interests are nonlocal and topological effects in quantum mechanics, quantum field theories and interpretations of quantum mechanics. In 1959, he and David Bohm proposed the Aharonov-Bohm Effect for which he co- received the 1998 Wolf Prize. http://en.wikipedia.org/wiki/Yakir_Aharonov ______

Gauge transformation Aharonov-Bohm effect Young's double slits experiment Vector potential

______1. Gauge transformations in electromagnetism We start with the Maxwell's equations,

 E  4  1 B   E    c t  B  0  4 1 E   B  j   c c t where

B: magnetic field E: electric field : charge density J: current density

The Lorentz force is defined as

1 F  q[E  (v  B)] . c

The Lorentz force is expressed in terms of fields E and B, which is invariant under the gauge transformation (gauge independent). The magnetic field B and electric field E can be expressed by

B    A ,

1 A E     , c t where A is a vector potential and  is a scalar potential. When E and B are given,  and A are not uniquely determined. If we have a set of possible values for the vector potential A and the scalar potential , we obtain other potentials A’ and ’ which describes the same electromagnetic field by the gauge transformation,

A', A  

1  ',   c t where  is an arbitrary function of r. We note that B and E are gauge-invariant;

B'   A'   (A  )    A  B

1 A' 1 (A  ) 1  1 A E'   '   (  )      E c t c t c t c t

2 Canonical momentum and mechanical momentum We now consider the Lagrangian which is defined by

1 1 L  mv 2  q(  v  A) , 2 c where m and q are the mass and charge of the particle. The Canonical momentum is defined as

L q p   mv  A . v c

The mechanical momentum  is given by

q π  mv  p  A. c

Then the Hamiltonian is obtained as

q 1 1 q H  p  v  L  (mv  A)  v  L  mv 2  q  ( p  A)2  q . c 2 2m c

The Hamiltonian formalism uses A and , and not E and B, directly. The result is that the description of the particle depends on the gauge chosen.

3. Change of the wave function under a gauge transformation (by Mathematica) The Schödinger equation contains the vector potential A. It may imply that the wave function may change as the vector potential A and scalar potential  is changed according to the gauge transformation,

1  A' A   , '    c t

The Schrödinger equation in the gauge (A,  ) takes the form

1 e 2  [ ( p  A)  e]  i  . 2m c t where , A and  depends on r and t. We note that the charge of electron is denoted as –e. The Schrödinger equation in another gauge ( A' ,' ) takes the form

1 e 2  [ ( p  A')  e'] ' i  ' 2m c t where ' , A' and  ' depends on r and t. The wave function changes as

ie  ' exp( ) , c under the gauge transformation. The difference between  ' and  is only the phase factor. We now give a proof for this by using the Mathematica. We show that

1 e e 2 e   [ ( p  A  )  e  ]exp(i)  i exp(i) 2m c c c t t is equivalent to

1 e 2  [ ( p  A)  e]  i  . 2m c t

First we assume that

 '  exp(i) where  is just a parameter to be determined. We will show that

e   . c

((Mathematica)) We assume that the electron charge is denoted by –e1 in the program, which means e1>0. (i) We need to calculate directly

1 e e 2 e   eq1  [ ( p  A  )  e  ]exp(i)  i exp(i) 2m c c c t t 4

in the Cartesian co-ordinates. This equation reduces to

1 e 2  eq2  exp(i){[ ( p  A)  e]  i } 2m c t

e with   . c

The calculation without Mathematica is too complicated for me.

((Program)) Here is a Mathematica program which I made. Clear "Global`" ; 1   x, y, z, t ;ex 1, 0, 0 ; 1 ey  0, 1, 0 ;ez 0, 0, 1 ; 1   x, y, z, t  D 1, t ; c   —     1   x, y, z, t ;p: Grad  , x, y, z &;          A1  Ax x, y, z, t , Ay x, y, z, t , Az x, y, z, t  e1 Grad 1, x, y, z ; 1x : ex. p    A1  &; c   e1    e1  1y : ey. p   A1  &; 1z : ez. p   A1  &;   c   c H1 : 1 1x 1x  1y 1y  1z 1z   e1 1  &; 2 m s1  H1 1 — D 1, t FullSimplify;           rule1    0 1, 2,3, 4Exp  1, 2, 3, 4 & ; rule2    0,   0 1, 2, 3, 4 & ; s2  s1 . rule1 FullSimplify;         

 

s3  s1 . rule2 FullSimplify; eq1   x,y,z,t s2  s3 FullSimplify;   eq2  Solve eq1  0,  1 e1   c —      x,y,z,t s2   . eq2 FullSimplify s3   1   s3 1  e12 Ax x, y, z, t 2 0 x, y, z, t  2c2 m e12 Ay x, y, z, t 2 0 x, y, z, t  2 2 2 e1 Az x, y, z, t 0 x, y, z, t  2c e1 m  x, y, z, t 0 x, y, z, t  2  c2 m — 0 0,0,0,1 x, y, z, t       ce1— 0 x, y, z, t Az 0,0,1,0 x, y, z, t          2  ce1— Az x, y, z, t 0 0,0,1,0 x, y, z, t        c2 —2 0 0,0,2,0 x, y, z, t        ce1— 0 x, y, z, t Ay 0,1,0,0 x, y, z, t        2  ce1— Ay x, y, z, t 0 0,1,0,0 x, y, z, t     c2 —2 0 0,2,0,0 x, y, z, t        ce1— 0 x, y, z, t Ax 1,0,0,0 x, y, z, t        2  ce1— Ax x, y, z, t 0 1,0,0,0 x, y, z, t     c2 —2 0 2,0,0,0 x, y, z, t     

4 Analogy from Classical mechanics   The Newton’s second law indicates that the position and the velocity take on, at every point, values independent of the gauge. Consequently,

r' r and v'  v , or

π' π ,

q Since π  mv  p  A, we have c

q q p', A' p  A c c or

q q p' p  (A'A)  p   . c c

In the Hamilton formalism, the value at each instant of the dynamical variables describing a given motion depends on the gauge chosen.

5. Gauge invariance in quantum mechanics In quantum mechanics, we describe the states in the old gauge and the new gauge as  and  ' . The analogue of the relation in the classical mechanics is thus given by the relations between average values.

 ' rˆ  '   rˆ  (gauge independent)

 ' πˆ  '   πˆ  (gauge independent) or equivalently

q q  'pˆ  A'  '   pˆ  A c c

(we will discuss the proof later). We now seek a unitary operator Uˆ which enables one to go from  to  ' .

 '  Uˆ 

From the condition  '  '    , we have

Uˆ Uˆ  UˆUˆ   1ˆ

From the condition,  ' rˆ '   rˆ

Uˆ  rˆUˆ  rˆ or

ˆ ˆ U [rˆ,U ]  0  i pˆ

Uˆ is independent of pˆ . We also get

q q Uˆ  ( pˆ  A')Uˆ  pˆ  A c c or

q q q Uˆ  ( pˆ  A  )Uˆ  pˆ  A c c c or

q q q Uˆ  pˆUˆ  Uˆ  AUˆ  Uˆ Uˆ  pˆ  A c c c q q q  A    pˆ  A c c c q    pˆ c

Note that [rˆ,Uˆ ]  0 , and A is a function of rˆ .

((Note)) Here we show that

 ' πˆ  '   πˆ  is equivalent to

q q  ',pˆ  A'  '   pˆ  A c c

q q where πˆ  pˆ  A and A'. A   c c

((Proof))

q q q  ' pˆ  A'  '   pˆ      Uˆ  A'Uˆ  c c c q q   pˆ      A'  c c q q   pˆ    A'  c c q   pˆ  (A') c q   pˆ  A c where [Uˆ, A']  0ˆ , since A ' is a function of rˆ .

6. Expression of the unitary operator We assert that Uˆ

iq iq Uˆ  exp[ (rˆ)] , Uˆ   exp[ (rˆ)]. c c

Then we get

Uˆ  pˆUˆ  Uˆ  [ pˆ,Uˆ ] pˆ

ˆ   ˆ  U i U  pˆ rˆ

ˆ  ˆ iq  U Ui   pˆ c q  pˆ   c which coincides with the expression described above.

((Note)) We use the notation such that

 iq (rˆ) iq Uˆ  Uˆ  Uˆ () . rˆ c rˆ c

______So we get the gauge transformation for the wave function;

iq  '  exp[ (rˆ)] . c

iq r  '  exp[ (r)] r  c

The phase factor of the wave function depends on the choice of the form of  in the gauge transformation. potential

7. Hamiltonian under the gauge transformation We consider the Schrödinger equation given by

 ˆ i   H  . t and

 ˆ i  '  H '  ' , t or  ˆ ˆ ˆ i U   H 'U  , t or

ˆ U ˆ  ˆ ˆ i[  U  ]  H 'U  . t t

Uˆ iq  Since  Uˆ , we get t c t

q  ˆ ˆ  ˆ ˆ  U  Ui   H 'U  , c t t or

q   Uˆ  UˆHˆ   Hˆ 'Uˆ  , c t or

q  Hˆ 'Uˆ   Uˆ UˆHˆ . c t

Thus we have

q  Hˆ '  UˆHˆUˆ  . c t

Note that

q Uˆ  pˆUˆ  pˆ   , c or

q pˆUˆ  Uˆ ( pˆ  ) , c or

q pˆUˆ  Uˆpˆ  Uˆ . c

We also note that

UˆpˆUˆ   Uˆ[ pˆ,Uˆ  ]  pˆ

ˆ  ˆ   Ui U  pˆ rˆ

ˆ ˆ  iq  UU i   pˆ c q  pˆ   c or

q UˆpˆUˆ   ( pˆ  ) . c

From the these relations, we get

q q q q Uˆ ( pˆ  A)Uˆ   ( pˆ  A  )  ( pˆ  A') , c c c c and

q q Uˆ ( pˆ  A)2Uˆ   ( pˆ  A')2 . c c

Then we have

q  1 q Hˆ '  Uˆ[ ( pˆ  A)2  q]Uˆ  c t 2m c or

q  1 q 1 q Hˆ '   ( pˆ  A')2  q  ( pˆ  A')2  q' . c t 2m c 2m c

Therefore the Schrödinger equation can be written in the same way in any gauge chosen.

8. Invariance of physical predictions under a gauge transformation The current density is invariant under the gauge transformation.

1 q J  Re[  pˆ  A ] m c

q q  ' pˆ  A'  '   Uˆ  ( pˆ  A')Uˆ  c c

q q q q Uˆ  ( pˆ  A')Uˆ  pˆ    A' pˆ  A c c c c

1 q 1 q J' Re[  ' pˆ  A'  ' ]  Re[  pˆ  A ]  J m c m c

Note: after the gauge transformation, A  A' in the current density operator. This is identified from the form of Hamiltonian.

1 q Hˆ ' ( pˆ  A')2  q' . 2m c

We note that the density is gauge invariant,

2 2 ' r  '    r  .

9. Aharonov-Bohm effect In the best known version, electrons are aimed so as to pass through two regions that are free of electromagnetic field, but which are separated from each other by a long

cylindrical solenoid (which contains magnetic field line), arriving at a detector screen behind. At no stage do the electrons encounter any non-zero field B.

Fig. Schematic diagram of the Aharonov-Bohm experiment. Electron beams are split into two paths that go to either a collection of lines of magnetic flux (achieved by means of a long solenoid). The beams are brought together at a screen, and the resulting quantum interference pattern depends upon the magnetic flux strength- despite the fact that the electrons only encounter a zero magnetic field. Path denoted by red (counterclockwise). Path denoted by blue (clockwise)

We assume that q = -e (e>0). In the space when B = 0, we have

B    A  0 , or

A   , or

r (r)   dr  A(r) , ro where r0 is an arbitrary initial point in the field region. We now consider the gauge transformation such that

A' A  ()  0 .

The new wavefunction  '(r ) can be written as

ie  '(r)  exp( ) (r) . c

The Schrödinger equation for  '(r ) is

2    2 ' i  ', 2m  t where  ' is the field-free wave function and the new Hamiltonian is that of free particle;

1 Hˆ ' pˆ 2 . 2m

Then we have

ie ie r   'exp( )  'exp[ dr  A(r)] , c c    ro

Reflector

Incident Electron beams slit-1 PQslit-2 Screen B out of page

Reflector

Fig. Schematic diagram of the Aharonov-Bohm experiment. Incident electron beams go into the two narrow slits (one beam denoted by blue arrow, and the other beam denoted by red arrow). The diffraction pattern is observed on the screen. The reflector plays a role of mirror for the optical experiment.

Let  1B be the wave function when only slit 1 is open.

ie  1,B (r)  1,0 (r)exp[ dr  A(r)], (1) Path1 c

The line integral runs from the source through slit 1 to r (screen). Similarly, for the wave function when only slit 2 is open, we have

ie  2,B (r)  2,0 (r)exp[ dr  A(r)], (2) Path2 c

The line integral runs from the source through slit 2 to r (screen). Superimposing Eqs.(1) and (2), we obtain

ie ie  B (r)  1,0 (r)exp[ dr  A(r)]  2,0 (r)exp[ dr  A(r)] Path1 Path2 c c

The relative phase of the two terms is

dr  A(r)  dr  A(r)  dr  A(r)  da (  A) Path1 Path2 

 da  B    B

by using the Stokes' theorem.  B is the magnetic flux. Then we have

ie ie  B (r)  exp[ dr  A(r)][ 1,0 (r)exp(  B )  2,0 (r)] , Path2 c c where the relative phase now is expressed in terms of the flux of the magnetic field through the closed path.

When

e e   da  B  2n (n =0, 1, 2, 3,…). B  c c Closed path

The pattern will be the same as without the magnetic field present.

When

e e 1   da  B  2 (n  ) , B  c c Closed 2 path or

2 c 1 1    (n  )  2 (n  ) , B e 2 0 2 the position of the minimum and the maximum in the pattern will be interchanged. 0 is the magnetic flux quanta and is given by

2 c    =2.067833667 x 10-7 Gauss cm2 = 2.067833667 x 10-15 T m2. 0 2e

((Note))

ikr1 ikr2 1,0 (r)  e ,  2,0 (r)  e

The condition for constructive interference in the presence of a magnetic field is

e kr1   B  kr2  2 c where l is intergers.

1 e r1  r2  ( B  2) . k c

The positions of the interference maxima are shifted due to the variation in  B , although the electron does not penetrate into the region of finite magnetic field. e (i) When  B  2n c

1 r  r  2 (n  ) . 1 2 k 

The pattern is the same as without B. e 1 (ii) When  B  2 (n  ) , c 2

1 1 r  r  2 (n   ) . 1 2 k  2

The pattern is different from that without B.

10. Young's double slit experiment for the Aharonov-Bohm effect

C1 slit-1

Electron beamsB Screen C2 slit-2

Fig. Young's double slit experiment with the electron beam source. The magnetic field is applied just behind the slits. There is no magnetic field around the paths C1 and C2.

e eBA B   B  . c c

r  r  d sin . 2 1

The phase difference;

2 2 2   r  r  d sin  d 0  2 1  

Since y  D tan  D , we have

2 d   y . 0  D

The intensity I

2   I = (r)  [1 ei(0 B ) ][1 ei(0 B ) ]  4sin 2 ( 0 B ) . B 2

The intensity is described by

2 d eBA y    I  4sin 2 ( 0 B )  4sin 2 (  D c ) . 2 2

When the effect of the width of the slit a is taken into account, the intensity is modified as

2 d eBA y  2   sin  I  4sin 2 ( 0 B )  4sin 2 (  D c ) , 2 2  2 where

a a a y   sin    .    D

1.0

0.8

0.6

0.4

0.2

-3 -2 -1 0 1 2 3 Fig. Young's double slits diffraction with Aharonov-Bohm effect. The diffraction pattern changes with the magnetic field. red (B = 0). Blue (B=intermediate value). Green (B = stronger field).

11. The observation of Aharonov-Bohm effect by Akira Tonomura

Fig. Picture of Dr Akira Tonomura (April 25, 1942– May 2, 2012), who was a Japanese physicist, best known for his development of electron holography and

his experimental verification of the Aharonov–Bohm effect. http://en.wikipedia.org/wiki/Akira_Tonomura

______Summary of the article (by A. Tonomura) [http://physicaplus.org.il/zope/home/en/1224031001/Tonomura_en]

(i) A toroidal ferromagnet (permalloy) instead of a straight solenoid, which has inevitable leakage fluxes from both ends of the solenoid. An ideal geometry with no flux leakage can be achieved by the finite system of a toroidal magnetic field. (ii) The toroidal ferromagnet is covered with a superconducting niobium layer to completely confine the magnetic field. (iii) An electron wave is incident to a tiny toroidal sample fabricated using lithography techniques. (iv) The relative phase shift between two waves passing through the hole and around the toroid is measured as an interferogram. A relative phase shift of π is produced, indicating the existence of the AB effect even when the magnetic fields are confined within the superconductor and shielded from the electron wave. An electron wave must be physically influenced by the vector potentials. Therefore, it can be concluded that electron waves passing through the field-free regions inside and outside the toroidal magnet are phase-shifted by π, although the waves never touch the magnetic fields.

Fig. Schematic diagram of the Aharonov-Bohm effect (by A. Tonomura group at Hitachi).

Fig. The direction of the vector potential around the solenoid coil.

Vector potential around the solenoid

   B  da  (  A)  da   A dl

Then we have

 A 2,r   or A    2r where the vector potential A is in a two dimensional plane perpendicular to the axis of solenoid. A  A e . e is the unit vector along the tangential direction.

Fig. The vector potential distribution around the solenoid.

http://prlo.aps.org/files/focus/v28/st4/ABeffect_BIG.jpg

Look Ma, no fields. Electrons passing around opposite sides of an electromagnet feel negligible magnetic fields (purple), but the electromagnetic potential (green circles and arrows) affects them in opposite ways, leading to measurable consequences. Before the effect was proposed in 1959, physicists thought fields must interact directly with particles to cause measurable electromagnetic effects.

12. Flux quantization in superconductors The electrons form a Cooper pairs in superconductors. The wavefunction of the

Cooper pairs in the absence of a field is given by  0 (r ) . Then in a presence of a field, it becomes

2ie r  (r)  exp[ ds  A(s)] (r) B c  0  r0

A closed path (C) about the cylinder starting at the point r0 gives

2ie  (r )  (r)  exp[ ds  A] (r ) B 0 0  0 0 c C

Since the wavefunction should be single valued, we must have

2ie exp[  ds  A]  1 c C or

2e  ds  A  2n c C or

2n c hc   ds  A    n B  C 2e 2e where we use the Stokes theorem,

ds  A  da   A  da  B      B C A A

where  B is the total magnetic flux. Then quantized magnetic flux is given by

2c hc     = 2.06783372 × 10-7 Gauss cm2 0 2e 2e

REFERENCES J.J. Sakurai, Modern Quantum Mechanics, Revised Edition (Addison-Wesley, Reading, MA, 1994). Y. Aharonov and D.J. Bohm, Phys. Rev. 115, 485 (1959). R. Penrose: The Road to Reality, A Complete Guide to the Laws of the Universe (Vintage Book, Nerw York, 2004). A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano, and H. Yamada, Phys. Rev. Letts. Letts. 56, 792 (1986). R. Shankar, Principle of Quantum Mechanics, 2nd-edition (Kluwer Academic/Plenum Publisher, New York, 1994). F. Schwabl, Quantum Mechanics, 4-th edition (Springer Verlag, Berlin, 2007).

Mathematica demonstration http://demonstrations.wolfram.com/AharonovBohmEffect/

APPENDIX Magnetic field distribution

-2

-4

-6

-6 -4 -2 0 2 4 6

Fig. Magnetic field lines of the system of 7 current loops stacked along the z axis which are equally spaced. When the coil is wound tightly and there are more loops, the magnetic field inside become larger and more uniform. The magnetic field B forms a closed loop.

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