The Aharonov-bohm Effect In Relativistic Quantum Theory

The Aharonov-Bohm Effect in Relativistic Quantum Theory. Craig Millar 200405021 12521 Research Project Abstract In this report I will be introducing Relativistic Quantum Mechanics in order to lead to a discussion on the Klein paradox and the Aharonov-Bohm effect. Along the way will be a reminder of the Schrödinger equation, before moving onto the beginnings of relativistic quantum mechanics with the Klein-Gordon equation then onto the Dirac equation. Gauge theory plays a crucial part in the prediction of the Aharonov-Bohm effect, hence this will be introduced.

“I think I can safely say that nobody understands Quantum Mechanics.” – R P Feynman

Contents Abstract ................................................................................................................................................... 1 Contents .................................................................................................................................................. 2 Introduction ............................................................................................................................................ 3 Review of Non-Relativistic Quantum Mechanics .................................................................................... 3 The Schrödinger Equation ................................................................................................................... 3 Observables and Operators (3) ............................................................................................................. 4 Relativistic Quantum Mechanics ............................................................................................................. 7 The Klein-Gordon Equation ................................................................................................................. 7 Derivation ........................................................................................................................................ 7 Problems with the Klein-Gordon Equation ..................................................................................... 8 Recovering the Schrödinger Equation from the Klein-Gordon Equation ...................................... 10 Dirac Equation ................................................................................................................................... 11 Energy-Momentum Relationship .................................................................................................. 12 Probability Density ........................................................................................................................ 13 Lorentz Covariance ........................................................................................................................ 14 Klein Paradox................................................................................................................................. 14 E-M Theory and Potentials .................................................................................................................... 19 Gauge Theory .................................................................................................................................... 19 Minimal Coupling .............................................................................................................................. 21 Gauge Transform of the Dirac Equation ........................................................................................... 23 Aharonov-Bohm (A-B) Effect ................................................................................................................. 24 Conclusion ............................................................................................................................................. 29 Acknowledgements ............................................................................................................................... 29 References............................................................................................................................................. 30

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Introduction The Aharonov-Bohm effect was first indicated by Ehrenberg and Siday in their paper on the refractive index in electron optics in 1948(1) but little attention was paid to it at the time. The phenomenon was rediscovered independently in 1959 by Aharonov and Bohm (2). In brief the theory says that in quantum mechanics potentials can act on charged particles even where the field lines are excluded, this will be discussed in detail in the last part of the report. This report begins at the beginning with a simple reminder of the Schrödinger equation in Non-Relativistic Quantum Mechanics (N.R.Q.M.) and operators and observables.

We shall then follow the

historical route to the Dirac Equation for Relativistic Quantum Mechanics (R.Q.M.) via the KleinGordon equation pointing out its flaws along the way. Once we have derived the Dirac Equation we will look at the Klein Paradox in which we are in the odd situation where you seem to get out a larger current than you put in from an electron incident on a potential step. We will then look at Gauge theory and the coupling of the Dirac equation to the electromagnetic field and proving that the Dirac equation is invariant under a gauge transform. Once we have covered all of these points we are in a position to discuss the Aharonov-Bohm effect in Quantum Mechanics in detail.

Review of Non-Relativistic Quantum Mechanics The Schrödinger Equation The Schrodinger equation allows us to predict the probability of finding a particle at a specific point in time or space so long as we know its wave function, . The Schrodinger equation can be justified as a suitable equation by taking a simple plane wave and using Planck’s law and the de Broglie’s hypothesis. (3) If we take the standard wave equation for a plane wave to be

  A exp  i  k  x  t  

(1.1)

Where A is the complex normalisation constant, k is the wave vector, x is the position vector, ω is the angular frequency and t is time. Planck’s Law is E   and the de Broglie hypothesis states that p  k . For the moment it is assumed that we are working in one dimension. Taking the 2nd derivative of (1.1) with respect to x we get

 2  k 2 x 2

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(1.2)

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Taking the kinetic energy and multiplying by (1.1) and then using the de Broglie hypothesis we get

E 

p2 2 k 2   2m 2m

(1.3)

Now replacing k 2 with (1.2), we are lead to

 2  2 E   2m x 2

(1.4)

Now the derivative of (1.1) with respect to time is taken.

  i t  

i   dt

(1.5)

Multiplying Planck’s Law by the wave function

E  

(1.6)

 t

(1.7)

Now inserting (1.5) into (1.6)

E  i

Since E must be the same in each case (1.4) and (1.7) can be set equal to each other to get

  2  2 i  t 2m x 2

(1.8)

This is the Schrödinger equation for a free particle. The Schrödinger equation can easily be changed to three dimensions and also changed for a particle in a potential to form the time dependant Schrödinger equation below.

 2 2 i     V  x, t  t 2m 2 Where  

(1.9)

2 2 2   x 2 y 2 z 2

Observables and Operators (3) In Quantum mechanics physical quantities such as energy, position and momentum are referred to as observables. Each observable has an operator associated with it to allow it to be calculated Craig Millar 200405021

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from the wave function. An observable usually represented by A will be represented by the operator Aˆ and the expectation values and mean square can be calculated by 





Aˆ    * ( x, t ) Aˆ  x, t  dx 







Aˆ 2    * ( x, t ) Aˆ 2  x, t  dx 

(1.10)

(1.11)

Taking a generalised wave equation relating the position wave function, , and the momentum wave function,  , through a Fourier transform we have (1.12) in one dimension. 

1  ipx    x, t     p, t  exp  dp  2     

(1.12)

The inverse of equation (1.12) is given by 

1  ipx    p, t     x, t  exp  dp  2     

(1.13)

In both equations (1.12) and (1.13), p is the momentum and the rest of the symbols have the same definitions as in the Schrödinger equation. Calculating the expectation values of the momentum in the usual way 

p 

 p   p, t 

2

dp

(1.14)



Where  ( p, t )   * 2

If we multiply p into both sides of equation (1.12) we get

p  x, t  

1 2 



 ipx  dp  

 p  p, t  exp 





 1  ipx   i   p, t  exp  dp  x 2        i   x, t  x

(1.15)

So now the expectation value of p becomes

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

   p    *  x, t   i   x, t  dx x   

(1.16)

By taking a similar approach with the expectation value of the mean square of the momentum

p 2  x, t  

1 2 

 i



 ipx  dp  

 p   p, t  exp  2



 1 x 2 



 ipx  dp  

 p  p, t  exp 



   1  i  i x  x 2    2



 ipx   dp     

   p, t  exp 



   x, t  x 2 2

(1.17)

This leads to the expectation value of    2 p 2    *  x, t   2 2   x, t   dx x   

(1.18)

So from(1.10), (1.11) and (1.16), (1.18) we can see that the momentum observable , p, is represented by the momentum operator

 x

(1.19)

pˆ  i

(1.20)

pˆ  i Which becomes

In 3 dimensions. There is also an operator associated with the energy.

E

The energy observable is given by

p2  V  x  . To change this into an operator we have to put in the momentum operator 2m

(1.19) and the position operator x  xˆ giving

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pˆ 2 ˆ H  V  xˆ , t  2m     i  x    V  x, t  2m  2  2   V  x, t  2m x 2 2

(1.21)

Hˆ is known as the Hamiltonian operator and the eigenvalues of it are the energy of the system such that Hˆ  x, t   E  x, t  .(4)

Relativistic Quantum Mechanics The Schrödinger equation is not invariant under a general Lorentz transformation, i.e. each side of the equation transforms differently under a Lorentz transform. Therefore it is only useful for describing phenomena for velocities very much less than the speed of light. This is where we need to start using Relativistic Quantum Mechanics. The first attempt at a relativistic quantum theory was in 1926(5) by Klein and Gordon with the Klein-Gordon Equation but there were problems with their equation some of which will be discussed below. This was then followed in 1928 by the Dirac Equation which correctly describes elementary spin ½ particles, such as the electron. (6)(7)

The Klein-Gordon Equation Derivation (8)(9) In 1926 Klein and Gordon attempted a relativistic version of the Schrödinger equation by first finding a Lorenz invariant version of the Hamiltonian(1.21). They used

H 2  p 2 c 2  m2 c 4

(2.1)

H 2    2 2 c 2  m 2 c 4

(2.2)

Using (1.20), (2.1) becomes

When we return to the free particle Schrödinger equation with our new Hamiltonian we see that

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1/2       2 c 2 2  m 2 c 4   t 2  2 2     2 c 2 2  m2 c 4  t

i

(2.3)

Rearranging (2.3) we are lead to

 2    2 c 2 2  m 2 c 4  0 2 t 2 2    2 2    m 2 c 2  0  2 c 2 t 2 x  2

 2

  x0 

2



 2

  x 

2

m2c 2  2  0 

 2 m2c 2     2   0    Which is the Klein-Gordon Equation, where 2 

(2.4)

1 2  2 . c 2 t 2

Problems with the Klein-Gordon Equation The first and most obvious problem with the Klein-Gordon equation is that when we try to find the energy from equation (2.1) we see that we have introduced a negative energy term such that

H

p 2c 2  m2c 4 OR  p 2c 2  m2c 4

(2.5)

The second problem with the Klein-Gordon equation is that of a positive definite probability density, ρ. In order to find the probability density we have to find the conservation equation for the Klein-Gordon equation. For this we need the complex conjugate of equation (2.4)

 2 m2c 2  *    2   0   

(2.6)

Taking equation (2.6) back a few steps in its derivation we return to

1  2 *  2 * m2c 2 *    0 c 2 t 2   x  2 2

(2.7)

If we now take the complex conjugate wave function multiplied by (2.4) and subtract the wave function times (2.7) and set it equal to 0. Craig Millar 200405021

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  1 2 2 m2 c 2      2    c 2 t 2   x  2     *

  2 2 m2 c 2  * 1     2  0  c 2 t 2   x  2    

(2.8)

This can be re-written as

  i  *   *   i *        *   0         2   t  2mc  t t   2m 

(2.9)

After some manipulation(8). If we then compare (2.9) with the standard continuity equation

   j  0 t

(2.10)

Where  is the probability density and j is the probability density current, we would like to interpret

i  *   *     2mc 2  t t 

(2.11)

as the probability density. However a probability density has to be a positive definite expression which this is not. To find out what (2.11) is, we use a plane wave of the form

  expi  k  x  t 

(2.12)

And

  2  i   2 t t 2   2  ik  k 2 2 x x and appropriate complex conjugates

(2.13)

Substituting (2.12)and (2.13) into (2.11) we get

i   i k  x t  i  k  x t   i  k  x t   i  k  x t    e   i   e  e  i  e 2   2mc



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(2.14)

i  i *  i *  2  2mc

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

i  2i *  2  2mc



  *  mc 2 

(2.15)

Which is an energy density since E   and E  mc 2 so we have a ratio or energies multiplied by a probability density giving us an energy density. Using equations (2.4) and (2.13) to find the allowed frequencies of (2.12) we find that

1  2 m2c 2 2      0 c 2 t 2 2 2 m2c 2   2   k 2  2   0 c  2 4 mc   2  c2k 2  2 

(2.16)

If we take the non-relativistic limit where   k ,equation (2.16) becomes



mc 2 

(2.17)

If we now substitute (2.17)into (2.15) we can recover the probability density

   *

(2.18)

for the Schrödinger equation as we would expect. Recovering the Schrödinger Equation from the Klein-Gordon Equation If the Klein-Gordon equation is to be a candidate description for relativistic quantum mechanics it would be sensible for it to collapse down to the Schrödinger equation in the non-relativistic limit. As we have already seen from (2.18) the probability density collapses to the correct form so what about the rest of the equation? Sticking with our plane wave

  expi  k  x  t 

(2.19)

And taking the first derivative with respect to time we get

  i t

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(2.20)

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Taking the square root of (2.16) and rearranging 1

mc 2  k 2  2  2  1     m2c 2 

(2.21)

Now using a Taylor expansion to the first two terms to approximate (2.21)



1  2 2  mc 2  k  1  2  2 2     1!  m c    

mc 2 1 k 2     2 m

(2.22)

Substituting (2.22)into (2.20)

 mc 2 1 k 2    i    t 2 m     1 2k 2  mc 2   t 2 m   2  2  i   mc 2 2 t 2m x  i

(2.23)

Which we have already seen is the Schrödinger equation with a potential V  mc 2 .

Dirac Equation Much of the following is adapted from Bjorken and Drell. (8) In 1928 Dirac took the search for a relativistically covariant version of the Schrödinger equation down a different route to Klein and Gordon. He noticed that the Schrödinger equation is linear in the time derivative and decided that the Hamiltonian should be linear in the space derivatives too. He suggested an equation of the form

i

       ic  1 1   2 2   3 3    mc 2  H t x x   x

(3.1)

For this to be a suitable form the constants  and  must be square matrices and  must be a column matrix, this will led to a set of coupled first order equations

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i

N N         ic  1 1   2 2   3 3       mc 2  t x x x    

(3.2)

N

  H   1

In order for it to be a candidate it must satisfy the same energy momentum relationship as the Klein-Gordon Equation, give a continuity equation with a positive definite probability density and be Lorentz Covariant. Energy-Momentum Relationship In order for (3.1) to be a suitable equation it must first give the correct energy momentum relationship (2.3) as in the Klein-Gordon equation. This can be done by taking the derivative of (3.1) with respect to time and multiplying by i which gives

 2      i  i  2 t t  t 

 2 3

  2 c 2 

i , j 1

 j i   i j  2 2

x x i

3

j

 imc3   i    i  i , j 1

   2 m 2c 4 xi

(3.3)

In order for (3.3) to be equal to (2.3) the following must be true about the coefficient matrices

 i j   j i  2 ij  i    i  0

(3.4)

 i2   2  1 Where  ij  1 for i  j and 0 for i  j . The other restriction on the matrices is that they must be hermitian so that the Hamiltonian of equation (3.1)is also hermitian. From the third equation in (3.4) we know that the eigenvalues of the matrices are 1 and from the anti-commutation properties the sum of the diagonal elements, the trace, must be zero. From this we know that the matrices must also be even dimensional matrices because if trace is zero and the diagonal elements are the eigenvalues then there must be an equal number of positive and negative eigenvalues which can only occur if the matrix is evenly dimensional. The first suitable matrix is a 4x4 matrix, this is because a 2x2 matrix can only contain the 3 Pauli matrices because they are anti-commutating, and the identity matrix which can commute with the Pauli matrices which is not allowed by (3.4). The usual choice of the first matrix combination is

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0

i    i

i 

1

0

    0 1

 0

(3.5)

Where  i are the Pauli matrices,

0 1

1   x    1 0  0 i   0

2   y   i

1

(3.6)

0

3   z     0 1  1 is the 2x2 identity matrix and 0 is the 2x2 zero matrix. By using equations(3.4), (3.5)and (3.6) we can recover the correct energy-momentum relationship,(2.3) from (3.3). Probability Density The second criterion which (3.1) must satisfy is to give a positive definite probability density. To do this we need the hermitian conjugates of the wave function and of(3.1). The hermitian conjugate is the complex conjugate of the transpose of the matrix denoted by † . The hermitian conjugate of  is  † and the hermitian conjugate of (3.1) is

i

  †  †  †  †   ic  1† 1   2† 2   3† 3    † mc 2 † t x x x  

(3.7)

We know that i  i† and    † so (3.7) can be simplified. Left multiplying (3.1) by  † and subtracting (3.7)right multiplied by  gives

i †

3 3      †    †   i    ic  † k k  mc 2 †    ic  k  k  mc 2 †  t  t  x k 1  k 1 x  3    i  †  ic  k  † k  t k 1 x 3   (3.8)  i  †  ic  k  † k   0 t k 1 x

Comparing this with equation (2.10) we can make the identification that

   †

(3.9)

Which is what we would expect for a probability density. Craig Millar 200405021

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Lorentz Covariance First off we need to write the Dirac equation in a covariant form. To do this we rearrange (3.1) to equal zero then multiply by

 c

to get

      i   1 1   2 2   3 3    2 mc  0 x x x   c t       i   1 1   2 2   3 3   mc  0 x x x   c t

(3.10)

Since  2  1 from the last equation in (3.4).

We know that

1    0 and if we set  0   and  i   i where i  1, 2,3 we get to c t x

      i   0 0   1 1   2 2   3 3   mc  0 x x x   x      i   mc  0 x   

(3.11)

In this representation there is a slight change to the matrices in (3.5) to

 0

i    i

i   0

1

0

0    0 1

(3.12)

Equation (3.11) is Lorentz covariant and this is proved in detail in Bjorken and Drell chapter 2. (8) Klein Paradox Let’s have a look at what happens to an electron incident on a potential barrier in the z direction. For and electron in region 1 with an energy E and a momentum p=pz we have 2

E 2 2 2    p m c c

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(3.13)

Page 14

Figure 1 - Diagram of an electron incident on a potential barrier.

(10)

However in region 2 where it is now in a constant potential V0 equation (3.13) becomes 2  E  V0  2 2     p   m c c   2

(3.14)

Where p’ is the momentum of the electron in region 2. As we are only working in one dimension we can reduce the Dirac equation (3.1) to

i

    ic   3 t z 

 2    mc  

  c pˆ   mc 2   E

(3.15)

For region 1 with α=α3 and p=pz and for region 2 we get

 c pˆ   mc    E  V  2

0

(3.16)

If we now look at the possible solutions for the incident, reflected and transmitted waves. The incident wave in region 1, travelling left to right has a solution of (8)

 inc

 1   0     aeik1z  ck1   2  E  mc   0 

(3.17)

Remembering that p  k1 .

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The reflected wave in region 1 travelling right to left has a solution of

 ref

1    0     1  0      ik z  be  ik1z  ck1   be 1  0     2  E  mc   ck1  0    E  mc 2 

(3.18)

The wave transmitted into region 2 is subject to a constant potential so E  mc 2  E  V0  mc 2 giving

 trans

1 0         0 1       d eik2 z    deik2 z  ck2  0    2  E  V0  mc   ck2      E  V0  mc 2  0

(3.19)

In (3.19) p  k2 and the wave is again travelling left to right. In order for the solutions above to be correct the wave must be continuous across the boundary so that means that

 inc  ref   trans

(3.20)

By choosing a coordinate system where z=0 exactly at the boundary and using equations (3.17), (3.18), (3.19) and (3.20)we can calculate a, b, b’, d and d’ for the wave functions. Using the matrices we get 4 simultaneous equations.

aeik1z  be  ik1z  deik2 z

aeik1z

ck1  be ik1z E  mc 2 be ik1z

be  ik1z  d eik2 z ck1 ck2   eik2 z 2 E  mc E  V0  mc 2 ck1 ck2   d eik2 z 2 E  mc E  V0  mc 2

Now using the fact that at the boundary eikz  e0  1and letting r 

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(3.21)

k2 E  mc 2 k1 E  V0  mc 2

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ab  d b  d  a  b  rd b   rd 

(3.22)

From the 2nd and 4th equations in (3.22) we can see that the wave function is only continuous if

b  d   0 . This also tells us that there is no spin flip at the boundary. Solving the simultaneous equations in (3.22) we get

d 1  r  2 d b  1  r  2 b 1  r   a 1  r  a

(3.23)

d 2  a 1  r  These will be useful in a moment once we have calculated the particle currents. The particle current is given by j ( z )  c † ( z )3 ( z ) . This has to be calculated for the incident, reflected and transmitted waves. So

 jinc  ca †e  ik1z 1 0 

  ca † a 1 0 

ck1 E  mc 2

ck1 E  mc 2

 1  1 0  0   0 1 ik1z  ae  ck1  0 0  2   E  mc  0 0  0 

 ck1   E  mc 2    0  0   1     0 

ck1   ck1  ca † a    2 E  mc 2   E  mc c 2 k1  2a † a E  mc 2

Craig Millar 200405021

0 0   0 0 0   1 0  0 1

(3.24)

Page 17

Completing the same for the reflected and transmitted currents we get

c 2 k1 E  mc 2

(3.25)

c 2 k1 E  V0  mc 2

(3.26)

jref  2b†b and

jtrans  2d † d

Now if we compute the ratio of the reflected and transmitted currents to the incident current and using the equations in (3.23) we get

jtrans 4r  2 jinc 1  r 

(3.27)

1  r  2 1  r 

(3.28)

And

jref jinc Remembering that r 

2



k2 E  mc 2 . Looking at the limits of equations (3.27)and (3.28) as k1 E  V0  mc 2

V0  E  mc 2 , (i.e. r  0 ). Looking at (3.27) if r  0 then the whole equation is negative which is implying that the transmitted current is travelling in the opposite direction to the incident current. This is rather counter intuitive as you would expect the transmitted current to be travelling the same direction. Leaving (3.27) for a moment to look at(3.28). When r  0 the equation is negative, this implies that the reflected current is travelling in the opposite direction to the incident wave as expected but

jref jinc

 1 which implies that the reflected current is greater than the incident current which is

counter intuitive. Together the above analysis point to a current being produced in region 2 travelling from right to left. But where is this current coming from? We have assumed up until this point that region 2 is empty, clearly this cannot be the case as electrons are leaving it producing the current. So in order to make sense of this we have to reassess this assumption. If we change the assumption to be that all electron states with energy E  mc 2 must all be occupied with electrons, then in region 2 when a potential is applied the electron energy is raised. Craig Millar 200405021

Page 18

If the energy is raised enough there will be an overlap between the filled negative continuum of region 2 and the positive continuum of region 1. This then allows electron coming in from the left to knock extra electrons out of the vacuum in region 2 increasing the reflected current and giving the negative transmitted current we see from equations (3.27) and (3.28). If we are knocking electrons out of region 2, what is left behind? Originally it could only be one thing, the proton but this didn’t fit with the physics so Dirac postulated the existence of a particle identical to the electron only with a positive charge. It took the discovery of the positron in 1932 by Carl D. Anderson(11) to complete the picture and prove Dirac’s postulation correct.

E-M Theory and Potentials (Note: In all the remaining sections   c  1 for simplicity)

Gauge Theory Gauge theories are a general class of quantum field theories which are used to describe elementary particles and their interactions. Electromagnetic (EM) theory closely relates the behaviour of the forces to the symmetry principle. The principle of symmetry in the real world is basically an operation in which the object looks the same before and after. This can be applied to the laws of physics too, if we can perform some mathematical operation on an equation and leave it looking as it did before the equation is said to be invariant under a transformation. A very important fact that we need to know is the difference between a global and local invariance. A global invariance is one in which a transformation occurs simultaneously throughout space and time and a local invariance is one in which different transforms are carried out at different spacetime points. Unfortunately many globally invariant theories, such as EM, are not locally invariant but by a clever choice of fields acting on a particle a local invariance can be restored to the system. To demonstrate this I will use Maxwell’s equations. Maxwell’s equations are(12)

 E  

 E  

(4.1)

B t

 B  0

 B  j 

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(4.2)

(4.3)

E t

(4.4)

Page 19

In Heavyside-Lorentz units where ε0 and μ0 are unity by choice of suitable units for the charge and current and also rationalising to remove 4π(12). Maxwell modified (4.4) to include the

E term otherwise, from the continuity equation, (2.10), t

the charge density would have to be constant in time. The extra term also allows the charge density to be locally conserved. As we already know the continuity equation tells us that charge must be locally conserved which also means that it is globally conserved. Why must it be locally conserved? In order to globally conserve it but not locally we would have to be able to transmit signals instantaneously across any distance which is not possible because it would violate special relativity. Generally E and B are replaced by functions of a vector potential A ( x) as follows(12)

B   A

(4.5)

And

E  V 

A t

(4.6)

Where V is the scalar potential and A is the3-vector potential. Using (4.5) and (4.6) we can see that (4.2)and (4.3)are satisfied straight away. A major point to note about A and V is that they are not unique for a certain value of B or E which means that they can be transformed while leaving E and B unchanged. So what are the transformations which allow this to be the case? Using equation (4.5)we can see that

A  A  A  

(4.7)

Where χ is an arbitrary function of r and t. Putting (4.7)into (4.5) we see

B    A     A       A       A

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(4.8)

Page 20

Since   0 . Now if we look at (4.6) it is not quite so simple. This time V will have to transform as well as A. Let us now look at inserting (4.7)into (4.6)to find the transformation of V.

A t   V    A    t A   V    t t   A     V    t  t 

E  V  

If we now compare (4.9) to (4.6) we see that in order for it to be invariant V  V   implies that V   V 

(4.9)

 which t

 . t

So (4.9) now becomes

   A  E    V    t t  t  A  V  t

(4.10)

 This can all be re-written in a compact form using the 4-vector potential, A  V , A (12). This

transforms as

A  A 

Where

 x

(4.11)

     ,   x  t .

Minimal Coupling So now, how is the Dirac equation connected to electromagnetism? Remember that the Dirac equation is

i

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  i     m t

(4.12)

Page 21

With   c  1 but (4.12) is not invariant under gauge transformation. In order to make it gauge invariant we must now look at the Gauge Principle.(13) If we start by considering a gauge transform which has the form

   eiq

(4.13)

Where q is charge and χ and ψ are functions of r and t. If we now take the gradient of (4.13)

     eiq   eiq    iq 

(4.14)

iq  We see that we have now gained the term e  iq  . The method of resolving this problem

is to introduce the field A into the gradient in such a way as to be able to cancel out this term. So if we introduce the field using

    iqA

(4.15)

Now we get

   iqA      iqA eiq

   eiq   iqAeiq  eiq    iq  A    

(4.16)

Then if we make it so that A  A   the theory is now invariant under the gauge transformation. Now if we do the same for the time derivative of (4.13)

   iq  e   t t  iq   iq e   eiq  t t     eiq   iq  t   t  

We have gained the term eiq  iq

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(4.17)

   , if we now introduce the scalar potential V such that so t 

Page 22

    iqV t t

(4.18)

now the time derivative of (4.13) becomes

    iq   iqV       iqV   e   t   t     eiq   iqV eiq t       eiq   iq  V     t     t In the same was as for the gradient we can make it that V  V  

(4.19)

 . So taking what we know t

from (4.15)and (4.18)and putting it into the Dirac equation, (4.12)we can get it in the gauge invariant form

  i   iqV   i     iqA     m  t      i  qV     i  qA    m  t   i     i  qA    m  qV  t

(4.20)

For an electron with q  e this then becomes

i

     i  eA   m  eV  t

(4.21)

This equation expresses the minimal coupling of the Dirac equation to an electromagnetic field when the Dirac particle is taken to be a point charge, in this case an electron.(8)

Gauge Transform of the Dirac Equation Now let us prove that the Dirac Equation in the form of (4.20)is gauge invariant. Taking

i

      i  qA    m  qV   t

(4.22)

As our new Dirac equation and by using the transforms

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A  A  

(4.23)

 t

(4.24)

V V  And

   eiq

(4.25)

Let’s now see if we can get back to equation (4.20). Putting (4.23), (4.24)and (4.25)into (4.22)we get

i

       eiq     i  q  A       m  q  V    eiq  t t    

     iq  iq  i t  q  V  t   e    i  q  A       m  e   

(4.26)

Now multiplying out and performing the derivatives we get

   iq   iq   i  qV  q  q  e   q  i  q A   q   m  e  t  t  t  

(4.27)

Through cancellation of terms we are led to

   iq iq   i  qV  e   i  q A   m  e  t 

(4.28)

Upon dividing through by the phase factor and taking out common factors we get

   i t  qV     i  qA    m    i t     i  qA    m  qV 

(4.29)

Which we can see is identical to the original equation (4.20) hence the Dirac Equation coupled to the electromagnetic field in this way is gauge invariant.

Aharonov-Bohm (A-B) Effect If we think back to classical physics for a moment we will remember that only E and B are physically significant and the vector potential, A, and the scalar potential, V, can be changed by a

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Page 24

gauge transform without altering the physics of E and B. However this doesn’t hold in quantum physics as we will see shortly. In their paper(2) Aharonov and Bohm suggested two different effects, the magnetic A-B effect and the electrical A-B effect, both are dependent on their respective gauge transforms. We shall first look at the magnetic A-B effect as this is the better known of the two. If we start with the same gauge transforms as in (4.23), (4.24)and (4.25) but with a slight notation change to introduce the phase factor and replacing V with φ, to avoid confusion with V for the potential in the electrical A-B effect, we get

A  A   ( r , t )

   

(4.30)

 t

(4.31)

and

    exp  i( r , t ) 

(4.32)

Where the phase factor ( r , t )  q ( r , t ) . If we now take the first case where A( r )   ( r ) 

1 ( r ) then from (4.5)we have q

B   A     1     q  0

(4.33)

Since S  0 for all values of S and we also get(14) b

( r )  q  A( r )  dr 

(4.34)

a

Which is a line integral between points a and b, so by putting (4.34)into (4.32) we get that(14)

 b      exp  iq  A( r )  dr    a 

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(4.35)

Page 25

This now suggests that even if B=0, A can be non-zero which implies from (4.35)that the particle undergoes a phase change despite there being no force acting on it.

Figure 2- Charged particle moving along a closed loop

If we now look at a charged particle which travels from a to b along a path then back to a along another path as in Figure 2. The phase change along each path will be (4.34) so the total phase change will be

  q



A( r )  dr   q

path1



A( r )  dr 

path 2

q  A( r )  dr 

(4.36)

loop

By Stokes theorem (12) (4.36) can be rewritten as

  q  A  dr loop

q



 A  dS



B  dS  q

enclosed surface

q

(4.37)

enclosed surface

Where  is the magnetic flux within the closed loop. So now we know from (4.37)that the phase change,  , around the loop is the charge times the magnetic flux, q . Obviously if there is no magnetic flux inside the loop there will be no phase change as expected but if a charged particle is moving in a region where A≠0 there will be a phase change even if B=0. Aharonov and Bohm suggest an experiment based on the Youngs slits experiment in order to test this interesting phenomenon, it is shown in Figure 3. The electron beam is split by a set of Youngs slits positioned at A, these beams then pass either side of a tight wound solenoid and the brought back together at F where an interference pattern will be observed. If the B field is changed this will have an

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effect on the interference pattern at F despite the fact the magnetic field is contained within the solenoid and the electrons are only passing through an area where B  0 . This effect was first shown in experiment by Chambers(15) in 1960 using a set up similar the one in Figure 3.

Figure 3 - Schematic experiment to demonstrate interference with a time-independent vector potential

(2)

More recently in 1986 Tonomura et. al.(16) used a toroidal magnet coated with super-conducting material, to confine the magnetic field by the Meissner effect, and electron holography to see the effect. Figure 4 shows one of their sets of results, you will see in (a) and (c) that the interference lines are in different places inside and outside of the toroid which proves that there is a relative phase shift between electrons passing through the toroid and those passing around it. Meanwhile in (b) there appears to be no shift, this is because the phase has shifted my 2n thus making it appear to have not moved. The white dashed lines indicate where we would expect the interference lines to be if the toroid had no effect.

Figure 4 - evidence of the A-B effect.

(16)

The second type of A-B effect is the Electric A-B effect. Starting from the same equations as for the magnetic A-B effect and following the same process but this time with b

(t )  q   dt

(4.38)

a

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Which leads us to having

  

  

    exp  iq   dt  loop

(4.39)

Aharonov and Bohm suggested an experiment as in Figure 5 where A, B, C, D, E are devices for splitting and redirecting beams, W1 and W2 are wave packets, M1 and M2 are cylindrical metal beams and F is the interference region. The idea behind this is to have wave packets travelling along the system and have M1 and M2 change in potential only when the wave packets are well inside them, this protects the wave packets from the electric field because they are acting as a Faraday Cage. This means that the wave packet is never exposed to the potential but there will still be a change the scalar potential around the loop which will cause the interference in (4.38).

Figure 5 - Schematic of an experiment to demonstrate interference with a time-dependent scalar potential.

(2)

This experiment was difficult to realise so an alternative was sought and eventually in 1998 Oudenaarden et. al.(17) showed the electric A-B effect in an experiment designed to show both the electric and magnetic A-B effect using metal rings with tunnelling junctions in them so that a well defined voltage can be applied to either side of the ring. The A-B effect was originally worked out for the non-relativistic case so how would it change when taken to the relativistic case? Well it doesn’t, so long as you use the fully relativistic form for the vector and scalar potentials. In both the non-relativistic and relativistic cases all we get is a phase change which does not affect the physics of the wave function in any way as we have seen from the gauge transforms earlier in the report.

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Conclusion The relativistic quantum world is very different from the macroscopic world we live in. It allows things to happen which seem impossible, from the Klein paradox where we appear to get a larger current out than we put in, to the Aharonov-Bohm effect where wave functions undergo a phase change despite the fact that there is no force field acting on them, the change is entirely due to the vector potential or the scalar potential. In this report I have led you from the Schrödinger equation, through the Klein-Gordon equation and the Dirac equation, including the Klein paradox, and eventually ending up at the Aharonov-Bohm effect via Gauge theory.

Acknowledgements First of all I must thank Prof. Barnett for all his help and patience throughout this project, even on the 3rd attempt at explaining something to me. I would also like to thank Dr Jeffers for his help whilst Prof. Barnett was unavailable; you saved me becoming overly stressed. I would also like to thank Laura and Nick for helping proof read my report and finding the mistakes I just couldn’t see. Lastly I would like to thank all my friends for keeping me sane when things just didn’t seem to be going right.

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References 1. Ehrenberg, W and Siday, RE. The Refractive Index in Electron Optics and the Principles of Dynamics. Proc Phys Soc B, Vol 68, Issue 8, P8-21 (1949) 2. Aharonov, B and Bohm D. Significance of Electromagnetic Potentials in the Quantum Theory. Phys Rev, Vol 115, Issue 3, P485-491 (1959) 3. Barnett, SM. 12.229 Quantum Mechanics Lecture Notes, University of Strathclyde (2005) 4. Zaarur, W, Peleg, Y and Pnini, R. Quantum Mechanics, Schaum’s Easy Outlines. McGraw Hill, (2006) 5. O’Conner, J J and Robertson, E F.

Klein, Oskar Biography, The MacTutor History of

Mathematics Archive. School of Mathematics and Statistics, University of St. Andrews. (2001) http://www-history.mcs.st-andrews.ac.uk/Biographies/Klein_Oskar.html

[cited:

4th

December 2008] 6. Dirac, PAM. The Quantum Theory of the Electron. Proceedings of the Royal Society, Vol A117, P610-624 (1928) 7. Dirac, PAM. The Quantum Theory of the Electron (part II). Proceedings of the Royal Society, Vol A118, P351-361 (1928) 8. Bjorken, JD and Drell, SD. Relativistic Quantum Mechanics. Primus Custom Publishing (McGraw Hill Companies, Inc) (1998) 9. Aitchison, IJR. Relativistic Quantum Mechanics. The Macmillan Press (1972) 10. Gingrich, DM. Klein Paradox for spin ½ Particles. http://www.phys.ualberta.ca/~gingrich/phys512/latex2html/node68.html [cited: 16th January 2009] 11. Anderson, CD. The Positive Electron. Phys Rev Vol 43, Issue 6, P491-494 (1933) 12. Aitchison, IJR and Hey, AJG. Gauge Theories in Particle Physics. IOP Publishing (1989) 13. Bowley, R and Coles, P. Theoretical Elementary Particle Physics. University of Nottingham, February 2000. http://www.nottingham.ac.uk/~ppzfrp/particle_physics/notes.pdf [cited: 30th January 2009]

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14. Ritchie, D. Laboratory.

Advanced Quantum Mechanics.

Semiconductor Physics Group, Cavendish

http://www.sp.phy.cam.ac.uk/~dar11/pdf/AQPLecture%209%202008.pdf [28th

January 2009] 15. Chambers, RG. Shift of an Electron Interference Pattern by Enclosed Magnetic Flux. Phys Rev Lett, Vol 5, Number 1, P3-5. (1960) 16. Tonomura, A, Osakabe, N, Matsuda, T, Kawasaki T, Endo, J, Yano, S and Yamanda H. Evidence for Aharonov-Bohm Effect with Magnetic Field Completely Shielded from Electron Wave. Phys Rev Lett, Vol 56, Number 8, P792-795. (1986) 17. Oudenaarden, A, Devoret, MH, Nazarov, YV and Mooij JE. Magneto-electric Aharonov-Bohm Effect in Metal Rings. Nature, Vol 391, P768-770 (1998)

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