University of LjubljanaFaculty of Mathematics and Physics

Department of Physics

Seminar Ia

Berry’s Phase

Author: Andrej LebanAdviser: prof. dr. Anton Ramsak

Ljubljana, April 2013

Abstract

In this paper I will treat Berry’s phase in Quantum mechanics. Starting with the Quan-tum Adiabatic theorem as a necessary prerequisite, I will first give an example of similarphenomena in classical mechanics, and then derive Berry’s geometric phase in Quantummechanics. This will be further illustrated on basic examples. Finally, I will explain theAharonov-Bohm effect both by invoking a Dirac phase factor and by treating it as an ex-ample of Berry’s phase.

CONTENTS 1

Contents

1 Introduction 1

2 The Adiabatic theorem 1

3 Nonholonomic processes 2

4 Geometric phase in quantum mechanics 34.1 Geometric phase in three dimensional parameter space . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Berry’s phase near a degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 Example: particle with spin in a varying magnetic field 75.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Electron in a slowly varying magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6 The Aharonov-Bohm effect 96.1 The Aharonov-Bohm interference experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.2 The Aharonov-Bohm effect as an example of Berry’s phase . . . . . . . . . . . . . . . . . . . . . 10

7 Conclusion 11

1 Introduction

Berry’s phase, more properly (but less commonly) called geometric phase, is a measurable phase acquired byquantum system undergoing a slow (i.e. adiabatic), cyclic change in its parameters. It is a non-trivial phaseand cannot be done away with, depending only on the path through parameter space. Named after Sir MichaelBerry, who published an article [1] about it in 1984, it was actually first discovered by S. Panchararnam in 1956studying polarized light passing through crystals. A similar result, of a wave-function acquiring a non-trivialphase factor in (what can be thought of as) undergoing a cyclic (though not necessarily adiabatic) change,is known as the Aharonov-Bohm effect, and will be treated in this article from Berry’s viewpoint. All thingsconsidered, Berry’s phase is a general effect and thus it is almost surprising, it took until 1984 to firmly generalizethis phenomenon.

2 The Adiabatic theorem

Before tackling the geometric phase factor, it is necessary to explain what is meant by adiabatic in the contextof Quantum mechanics.

In classical thermodynamics, adiabatic means no heat is exchanged between the system and its surroundings.Here, however, no appeal to heat is made.

For an intuitive illustration of an adiabatic process in this sense, we imagine a perfect pendulum inside abox. If we carry the box around in a very smooth and steady way, the pendulum will continue swinging in thesame (or one parallel to it) plane with the same amplitude. Generally, it is necessary for such a process, thatthe characteristic external time Te - in our case the time it took us to move the box - is much larger thanthe characteristic internal time Ti - the period of our perfect pendulum: Te >> Ti.

The quantum mechanical concept of an adiabatic process can be stated in form of a theorem, first publishedby Max Born and Vladimir Fock in 1928: [2] [3]

Theorem 1. For a Hamiltonian undergoing a gradual change from some initial Hi to some final Hf , if thesystem is in an eigenstate |n〉 corresponding to the nth (discrete, non-degenerate1) eigenvalue of Hi, it will becarried into the eigenstate |n〉 of Hf with its corresponding eigenvalue.

1The theorem can be expanded to degenerate eigenvalues, in which case the states corresponding to that eigenvalue can mixamong themselves[4]

3 NONHOLONOMIC PROCESSES 2

Figure 1: Change in the Hamiltonian from Hi to Hf [3]

As an illustration, consider a particle in the ground state of an infinite square well in one dimension, withdimension a:

ψi =

√2

asin(

π

ax)

If we gradually move the right wall to 2a, the particle will end up in the ground state of the new, expandedwell:

ψf =

√1

asin(

π

2ax)

However, if the wall is moved suddenly, the particle will not instantly adapt to the change in the Hamiltonianand the resulting state will remain ψi, which is not an eigenstate of the new Hamiltonian, as seen in the figurebelow:

Figure 2: Initial state of the particle, the state after an adiabatic change, the state after a sudden change.[3]

What needs to be kept in mind is the fact that the adiabatic theorem applies not only to small perturbations,but to quite large (total) changes in the Hamiltonian, as long as they occur gradually.

3 Nonholonomic processes

In the preceding chapter, I have already alluded to the image of a perfect pendulum inside a box. Now imaginewe carry (adiabatically) the box around a closed path from the North Pole to the Equator, partway round theEarth on the Equator, and finally back up to the Pole, as seen in the following figure.

Figure 3: An illustration of our path, with the pendulum swinging in indicated planes. [5]

4 GEOMETRIC PHASE IN QUANTUM MECHANICS 3

This is not a mere planar transport due to the surface curvature of the sphere.It is obvious that, after our transport, the pendulum is swinging in a plane, which makes an angle θ with

regards to the initial one. Incidentally, the solid angle Ω subtended by our path is equal to θ:

Ω =A

R2=

1

2

θ

2π4πR2/R2 = θ (1)

As it turns out, the angle which the new plane makes with the old one is independent of the shape of thepath; as long as the subtended solid angle Ω is the same the angle. One can see a practical example in Paris,which is naturally not on the Equator. There the original Foucault pendulum undergoes daily precessionrelative to the Earth:

Figure 4: The original Foucault pendulum.

For an explanation of this, we usually invoke Coriolis forces, in this case the component parallel to thesurface of the earth. Coriolis force is a pseudo force which arises when we transform to a rotating frame ofreference - The Earth. The force a swinging pendulum feels is equal to: FC = −2m −→ω Earth ×−→v pendulum. Thecross product is obviously dependent on our latitude; the final plane of oscillation is the net effect of the dailycontribution of the force.

Instead of the mechanical explanation, we now turn to a purely geometric one. Imagine the pendulumbeing carried from the North Pole down to Paris (which makes an angle θ0 with regards to the North Pole),round the Earth - which represents the daily rotation, and back up to to the Pole. The solid angle subtendedby such a closed path is:

Ω =

∫sinθdθdπ = 2π(−cosθ)|θ00 = 2π(1− cosθ0) (2)

The net daily phase is equal to the subtended solid angle. Since the daily phase of the earth itself is 2π, thesecond term signifies the daily precession of the pendulum. No appeal to pseudo forces was needed here, sincewe were in an inertial frame - a stationary sphere. The pendulum’s case is generalized by the mathematicalnotion of Parallel transport [6].

Generally, a system whose state does not return to the initial one despite the parameters of the Hamiltonianreturning to their initial values is called nonholonomic.[3] The parameters need not be spatial coordinates, asin the above example, but any set of continuous parameters in the Hamiltonian.

4 Geometric phase in quantum mechanics

We now turn to the quantum description of an adiabatic, nonholonomic process, which will be described as aslowly taking a closed path in parameter space - as opposed to retracing one’s steps - i.e. going forwards andthen backwards to the starting point.

Suppose the Hamiltonian is time-dependent only via a vector of parameters R(t). The time-dependentSchrodinger equation is:

H(R(t)) |ψ(t)〉 = i~∂

∂t|ψ(t)〉 (3)

4 GEOMETRIC PHASE IN QUANTUM MECHANICS 4

At any instant - meaning fixed t, there exist a solution to the eigenvalue equation:

H(R) |n(R)〉 = En(R) |n(R)〉 (4)

However, the adiabatic theorem tells us that, for a gradually changing H, the state |n(R(0))〉 will remainin the state |n(R(t))〉 whatever the value of t, at most picking up a phase factor, which doesn’t affect theeigenvalue equation. We can therefore express the state, which was in |n(R(0))〉 at t=0, at any given timeas[1]:

|ψ(t)〉 = |n(R(t))〉 exp(−i~

∫ t

0

E′n(R(t′))dt′) exp(iγn(t)) (5)

The second term is called dynamic phase factor, and can be viewed as a generalized factor −iEnt/~, whenthe energy of an eigenstate is time-dependent. Further down, I will be using the following shorthand for thedynamic phase:

θn(t) = −1

~

∫ t

0

En(t′)dt′ (6)

To explain the form of the dynamic phase factor, note that the time evolution of a state is formally expressedas |ψ, t〉 = U(t, 0) |ψ, 0〉.[7] The time evolution operator U can be written as:

U(t, 0) = T exp(−i~

∫ t

0

H(t′)dt′) (7)

Here T is the time-ordering operator which orders the Hamiltonians as they appear in time[7]. None of whichmatters here, though, since the adiabatic theorem keeps us in the same (time-dependent) eigenstate, with thesame (time-dependent) eigenvalue. Hence T does not change the state, and we can replace the operator H inthe integral with the eigenvalue En(t).

More importantly, the adiabatic theorem does not rule out an additional phase, which we call geometricphase - our third term in Equation 5. We cannot assume it depends on time only via the parameters in R,which, as it turns out, has interesting consequences.

Plugging our wave function (Eq. 5) into the Schrodinger equation i~∂ψ∂t = H(t)ψ, remembering that thewave function (Eq. 5) is an eigenfunction of the Hamiltonian at any instant in time,2, though, normally, bothchange with time.

Thus we obtain a differential equation for the geometric phase factor:

dγndt

= i〈n(R(t))|∂|n(R(t))〉∂t

〉 (8)

If the parameter vector R has only one component, integrating the previous equation will give:

γn(t) = i

∫ t

0

〈n(R(t))|∂|n(R(t))〉∂t

〉dRdt′

dt′ = i

∫ Rf

Ri

〈n(R)|∂|n(R)〉∂R

〉dR (9)

If the parameter returns to the initial value after our adiabatic process(ending at time T ): Rf = Ri, thenthe geometric phase vanishes: γn(T ) = 0! This is equal to the fact that you cannot make a closed loop in onedimension, the only way to arrive at the start again is by going backwards - retracing your steps. The pendulumcarried from the Pole to the Equator and back the same way acquires no phase.

However, if there are more components in R, the partial time derivative becomes a gradient in parameterspace:

∂|n(R(t))〉t

= (∇R|n(R(t))〉) · dRdt

(10)

If we again take a closed path in parameter space, we arrive at the final expression for the geometric(Berry’s) phase factor after adiabatically traversing a closed path:

2This only holds in the extreme adiabatic limit, where no admixture of other states in |ψ(t)〉 (Eq. 5) occurs. For more, see [3],page 338.

4 GEOMETRIC PHASE IN QUANTUM MECHANICS 5

γn(C) = i

∮C

〈n(R)|∇Rn(R)〉 · dR (11)

The bra-ket term above is often called the Berry potential, which will be further illustrated in the followingsubsection.

Some properties of the geometric phase:

• It only arises when there is more than one time-dependent parameter in the Hamiltonian.

• It is only dependent on the path taken, not the time (provided it being sufficiently long enough for theadiabatic approximation to hold).

• For it to be a true phase factor, γn(C) must be real. That means 〈n(R)|∇Rn(R)〉 must be imaginary.That can be shown to always hold 3, provided the eigenfunctions are non-trivially complex.

• It is measurable, for example splitting a beam in two and sending one part through an adiabaticallychanging potential, whilst the other is left alone. When we recombine both beams again, the beam whichunderwent the potential will have acquired an additional phase factor, of both dynamic and geometricorigin. Such a phase factor is measurable by the interference pattern in |ψ|2. The dynamic part’s contri-bution is known, and the experiment can be so arranged to minimize it [3]. A similar experiment was alsosuggested by Berry in the original paper [1].

4.1 Geometric phase in three dimensional parameter space

When the Hamiltonian depends on time via three parameters, we can use Stokes’ theorem to express Eq. 11in another way[1]:

γn(C) = −Im∫ ∫

C

(∇× 〈n|∇n〉) · dS (12)

Here an interesting analogy with the equation describing magnetic flux in terms of the magnetic vectorpotential arises:

Φ =

∮C

A · dr =

∫S

(∇×A) · dS (13)

The term i 〈n|∇n〉 plays the role of the magnetic vector potential, while the ”flux” signifies the geometricphase; as I will show in the treatment of the Aharonov-Bohm effect (section 6), such analogies are morethan coincidental.

For now we can further express Eq. 12:

γn(C) = −Im∫ ∫

C

dS · 〈∇n| × |∇n〉

and expressing the cross product in the basis of eigenvectors[1]:

γn(C) = −Im∫ ∫

C

dS ·∑m 6=n

〈∇n|m〉 × 〈m|∇n〉

The term 〈m|∇n〉 can be obtained from the eigenvalue equation:

〈m|∇n〉 =〈m|∇H|n〉En − Em

m 6= n (14)

Using the previous relation, we mark the sum as Vn:

3∇R〈n(R)|n(R)〉 = 0 = 〈∇Rn(R)|n(R) + 〈n(R)|∇Rn(R)〉 = 〈n(R)|∇Rn(R)〉∗ + 〈n(R)|∇Rn(R)〉 = 0, which makes the innerproduct purely imaginary .

4 GEOMETRIC PHASE IN QUANTUM MECHANICS 6

Vn(R) = Im∑m6=n

〈n(R)|∇RH(R)|m(R)〉 × 〈m(R)|∇RH(R)|n(R)〉(Em(R)− En(R))2

(15)

Finally, the geometric phase factor expressed as a surface integral of an area, enclosed by C :

γn(C) = −∫ ∫

C

dS ·Vn(R) (16)

Expressing the geometric phase as a surface integral, we see that a simple retracing of steps in parameterspace brings no net phase factor, since it encloses no area (or, in analogy with the Foucault pendulum of previoussection, subtends no solid angle).

Comparing Eq. 12 and Eq. 16, we see the term Vn is basically the curl of a vector, making it gauge-invariant, which ensures the uniqueness of the geometric phase factor γn(C) [1]. In other words, tacking anotherphase factor on the states |n(R)〉 (which is always allowed) will not change the geometric phase. It dependsonly on the parameter space and the path taken through it.

The electromagnetic analogy can be taken further, defining a field strength tensor Fn for higher dimensions.In three dimensions its components are equal to Vn(R). For a detailed view on the whole subject, see [8].

4.2 Berry’s phase near a degeneracy

Figure 5: First row: A degeneracy in 2D is enclosed in the first two examples, but not in the third. Secondrow: In 3D, a point-like degeneracy is not enclosed.

Suppose that our path C takes us near a point R* in parameter space, where two (or more) states aredegenerate: Em(R∗) ≈ En(R∗). Naturally, their contribution to Vn(R) (Eq. 15) will far outweigh any other.If we denote these two states |+ (R)〉 and | − (R)〉 and expand the Hamiltonian for R near R* [1]:

V+(R) = Im〈+(R)|∇RH(R*)| − (R)〉 × 〈−(R)|∇RH(R*)|+ (R)〉

(E+(R)− E−(R))2(17)

et vice versa for V-(R). We see that V-(R) = −V+(R), meaning γ−(C) = −γ+(C). The phase changessign near a degeneracy.

A more general result, via[1]:

γ±(C) = ∓1

2Ω(C) (18)

,where Ω is the solid angle, subtended by the curve C from the point of degeneracy.For real, symmetric Hamiltonians we may choose eigenstates to be purely real. This constrains the phase to

multiples of π, since the wave function must be single valued, at most reversing the sign, meaning Ω = ±2π ifC encloses the degeneracy, and 0 otherwise[1] [5]. ”Enclosing” is a topological concept, meaning that the loopcannot be smoothly deformed to avoid the degeneracy. In two dimensions, a point-like degeneracy will trapsome loops, while in three dimensions, a line is needed:

5 EXAMPLE: PARTICLE WITH SPIN IN A VARYING MAGNETIC FIELD 7

Now we are moving into the related concept of topological phase since the solid angle our loop encloseshas become irrelevant. The final phase depends solely on the number of degenerate points enclosed, or related,the winding number of our path with regards to the point of degeneracy. Despite treating the Aharonov -Bohm effect as an example of Berry’s phase, it is more correctly a topological phenomenon, since it doesn’tmatter how large our loop is. More in ( Section 6 ).

5 Example: particle with spin in a varying magnetic field

5.1 General case

First, I’d like to show the spin reversal near a degeneracy on a general case of a particle with a spin in a magneticfield, following [1]:

Figure 6: A closed path in magnetic field parameter space, in this case with B of constant magnitude. Thepath encloses a solid angle Ω

The Hamiltonian for a particle with spin s interacting with a magnetic field (B) is[7]

H(B) = g~e2m

B · s (19)

The spin operator s has n = 2s+ 1 eigenvalues between −s and +s:

En(B) = g~e2m

Bn (20)

, with a n-fold degeneracy in case B = 0, which naturally reduces to the afore-mentioned twofold degeneracyin the case of a single electron. Note that the components of B take on the role of our parameter vector R.

Suppose we take an eigenstate |n, s(B)〉 of s in the direction of B, and slowly take it round a closed pathC in B as is illustrated in Figure 6. Practically, what I mean by that is the particle is kept stationary in realspace, while B is varied, eventually being brought back to its initial state. The adiabatic theorem ensures usthat the eigenstate will remain an eigenstate in the direction of B, however we will have to rotate our coordinatesystem as B changes direction.

Expressing Eq. 15 for this case, using the known Hamiltonian and eigenvalues, we get:

Vn(B) =Im

B2

∑m 6=n

〈n, s(B)|s|m, s(B)〉 × 〈m, s(B)|s|n, s(B)〉(m− n)2

(21)

First, we note that the only non-zero component of Vn is Vzn, the others containing terms with sz (followingthe rules of cross product) - 〈n|sz|m〉 which evaluate to zero, since |n〉 and |m〉 are both eigenfunctions of szand orthogonal to each other.

5 EXAMPLE: PARTICLE WITH SPIN IN A VARYING MAGNETIC FIELD 8

Using the rules for sx, sy, the only states that are coupled are those for which m = n±+1. The final resultfor Vzn is [1]:

Vzn =n

B2(22)

This gives us Vn(B) = [0, 0, Vz] with the z-axis always pointing in the direction of B, so in a stationarycoordinate system:

Vn(B) =nB

B3(23)

Using Eq. 16 for γn(C) we get:γn(C) = −nΩ(C) (24)

Where Ω is again the solid angle in parameter space subtended by the path C from the point of degeneracy- B = 0. This can be viewed as a quantum analogue to the transport of a perfect pendulum round the surfaceof the earth, acquiring a phase factor. For the case of an electron with n = 1

2 with B rotated through 2π in aplane, the phase factor is -1 - the wave-function (spinor) changes sign, as it encloses a degeneracy.

5.2 Electron in a slowly varying magnetic field

Now I wish to illustrate the previous result on a concrete case: an electron at the origin, subject to a magneticfield with constant magnitude, but changing direction - precessing round the z-axis with a constant ω, as shownin Figure 7. This example is a condensed version of the one found in [3].

Figure 7: Illustration of the precessing magnetic field in the case below [3].

We describe the field B as: B0[sinα cos(ωt), sinα sin(ωt), cos(α)]The Hamiltonian (Eq. 19), written in the Pauli spin matrices space, is:

H(t) = −~ωc2

(cosα e−iωtsinα

eiωtsinα −cosα

)(25)

where ωc is the familiar cyclotron frequency − eB0

m .Choosing the direction up to be in the whichever (instantaneous) direction of B (t), which is permitted by

the adiabatic assumption, the eigenspinors are:

χ+(t) =

(cosα/2

eiωtsinα/2

)χ−(t) =

(sinα/2

−eiωtcosα/2

)(26)

Such a system, starting in a pure ”up” state at t = 0, is exactly solvable [3], however, using the adiabaticassumption ω << ωc the time dependent solution is further simplified:

χ(t) ≈ χ+(t) eiωct/2 eiωcosαt/2 e−iωt/2 + i (ω

ωcsinα sin(

ωct

2)) e−iωt/2 χ−(t) (27)

6 THE AHARONOV-BOHM EFFECT 9

If we now assume perfect adiabacity ωωc→ 0, the solution only retains the first term and is basically the one

given in Eq. 5. The exponent with ωc is the dynamic phase, with corresponding energy eigenvalue E+ = −~ωc

2integrated from 0 to t. That leaves us with the geometric phase:

γ+(t) = (cosα− 1)ωt

2γ+(T = 2π/ω) = (cosα− 1)π

(28)

If α = π/2 , we get the familiar phase −π which reverses the sign of the wave-function (Eqs. 18, 24).

6 The Aharonov-Bohm effect

The Aharonov-Bohm effect deals with a charged particle acquiring an additional phase factor because of amagnetic field B (or, more specifically, its magnetic flux), despite being contained to a region where B = 0.However, the vector potential A is non-zero in the region, leading to a discussion whether electro-magneticpotentials are a basic physical entity.

It was first predicted by Ehrenberg and Siday in 1949, and published by Yakir Aharonov and David Bohmin 1959 [11]. First experimental confirmation came shortly thereafter by R. Chambers in 1960[9]. [10]

First, I am going to illustrate the Aharonov-Bohm interference experiment and derive the additional phasefactor by using a gauge transformation of the vector potential A, as is usually done in most elementary treat-ments of the effect, including [7] and [3], which will be my main sources. Later, I will show how the additionalphase factor can be treated as a Berry geometric factor.

6.1 The Aharonov-Bohm interference experiment

Consider an electron beam split in two, passing past an ideally infinitely long solenoid, but being barred from thevicinity of it. In addition to that, say we introduce an infinite potential barrier, preventing the wave-functionsfrom penetrating the vicinity of the coil, as is illustrated on Figure 8. Finally, we recombine both beams on ascreen.

Figure 8: Illustration of the Aharonov-Bohm interference experiment; the beams are recombined on a screen tothe right.[3]

Now, the magnetic field outside the coil is zero. That means:

B = ∇×A = 0 (29)

The vector potential A can always be transformed via a gauge transformation:

A→ A +∇Λ (30)

6 THE AHARONOV-BOHM EFFECT 10

where Λ is a scalar field.We know the curl of a gradient is zero, so it is convenient to express A as:

A = ∇Λ (31)

or

Λ(r) =

∫ r1

r0

A(s) · ds (32)

If we introduce a new gauge as in Eq. 30, the wave-function changes as [7]:

ψ′(x, t) = ψ exp(ie

~Λ(x, t) ) (33)

Instead of solving the Schrodinger equation for the region with A, we now use the solution for no vectorpotential, which is equivalent to introducing a new gauge transformation A’ = A +∇(−Λ).

If ψ′ is the solution for no A, the correct solution will be given by Eqs. 33, 34:

ψ = ψ′ exp(ie

~

∫ r1

r0

A(s) · ds ) (34)

We can now consider the aforementioned Aharonov-Bohm experiment. With only either of the slits open,the two wave-functions, sub-scripted 1 and 2 as in the illustration, will differ from their versions with no fieldpresent by:

ψ1/2,B = ψ1/2,0 exp(ie

~

∫1/2

A(s) · ds ) (35)

(here 1 and 2 also denotes the paths taken by the particles). If we open both slits, we get a linearsuperposition:

ψB = ψ1,0 exp(ie

~

∫1

A(s) · ds ) + ψ2,0 exp(ie

~

∫2

A(s) · ds ) (36)

The relative phase between both wave-functions is:∫1

A(s) · ds −∫2

A(s) · ds =

∮A(s) · ds =

∫∇×A(s) · dA = φmag (37)

The superposition, thus rewritten is:

ψB = (ψ1,0 exp(ie

~φmag ) + ψ2,0) exp(

ie

~

∫2

A(s) · ds ) (38)

The wave-function acquires a phase shift on account of φmag, despite not being in local contact with themagnetic field! This does not correlate easily with our idea of locality, unless we consider the potentials to bea basic physical entity, but then again, they are not gauge invariant. In either case, all observable quantitiescorrelate to B, not A - in this case the magnetic flux. This is a similar case to the one we already know fromthe Hamiltonian of a particle in electro-magnetic field, where only the potentials appear in the Hamiltonian,but all observable quantities are related to electro-magnetic fields.

6.2 The Aharonov-Bohm effect as an example of Berry’s phase

Berry devised a thought experiment, where charged particles are confined to a box at R outside a flux line,with the line not penetrating the box (Figure 9) [1].

The Hamiltonian and wave-functions will both exhibit a dependence on (r - R). If we again use the gaugetransformation trick, with zero-field eigenstates denoted by ψn(r−R), the states with magnetic flux present (inthe vicinity) will depend on the parameter R as:

〈r|n(R)〉 = ψn(r−R) exp (ie

~

∫ r

R

A(r′) · dr’ ) (39)

7 CONCLUSION 11

Figure 9: Berry’s thought experiment: the charged particles are confined to the box [1].

Again, this is only a phase factor. Now we transport the box round a circuit C as shown in the figure. UsingEq. 11 we get for the ”Berry potential”:

〈n(R)|∇Rn(R)〉 =

∫d3rψ∗n(r−R) [

−ie~

A(R)ψn(r−R) + ∇Rψn(r−R)] =−ie~

A(R) (40)

The integral replaces the sum for eigenfunctions with a continuous spectrum. The gradient ∇R is equal to −∇r,if acting on function of (r - R). This makes the second term the expected value of momentum of a stationarystate, which is zero.

The Berry phase is then:

γn(C) =e

~

∮C

A(R) · dR =e φmag

~(41)

Which is precisely the same result as in Eq. 38. Considering the phenomenon in topological terms, the centralflux line is itself a singularity for the vector potential, which can be expressed as A =

φmag

2πr eφ [3], hence thesize of our loop doesn’t matter.

7 Conclusion

The phenomenon of geometric phase is in no way limited to the examples described above. First of all, inclassical mechanics, there is an almost point-by-point analogy called Hannay’s angle, which I implicitly usedtreating the Foucault pendulum. For the derivation and more, see for example [12].

It also arises in polarized light, especially in interferometers, when a beam is split and then recombined,with one or both beams undergoing a path that is not planar [13]. This needs to be taken into account, and iscompletely analogous to the rotating spinor example.

The Born - Oppenheimer approximation[14] in molecular physics is itself an adiabatic approximation, withthe nuclear coordinates playing the role of the slow-changing external parameter (coordinates), and the electronicstates being instantaneous to them. In a special case, called the Jahn-Teller effect, Berry’s phase induces a signreversal of the wave-function, as discussed above. [8].

Due to the nature of this article, and also my own lack of knowledge, I have refrained from venturing too farinto topology, although there is a certain beauty in connecting mathematical concepts to physical phenomena.For an in-depth study, see [8].

REFERENCES 12

References

[1] Berry, M.V., Quantal Phase Factors Accompanying Adiabatic Changes, Proc. R. Soc. Lond. A. 392, 1984

[2] http://en.wikipedia.org/wiki/Adiabatic_theorem (8/4/2013)

[3] Griffiths, D.J., Introduction to Quantum Mechanics, Prentice Hall, 1995

[4] Rohrlich, D., Berry’s Phase, http://arxiv.org/abs/0708.3749

[5] http://www.mi.infm.it/manini/berryphase.html (8/4/2013)

[6] http://en.wikipedia.org/wiki/Parallel_transport (11/4/2013)

[7] Schwabl, F., Quantum Mechanics, Springer, 2007

[8] Bohm, A., Mostafazadeh, A., Koizumi, H., Niu, Q. and Zwanziger, J. The Geometric Phase in QuantumSystems, Springer, 2003

[9] Chambers, R., Shift of an Electron Interference Pattern by Enclosed Magnetic Flux, Phys. Rev. Lett. 5,1960

[10] http://en.wikipedia.org/wiki/Aharonov-Bohm_effect (11/4/2013)

[11] Aharonov, Y. and Bohm, D., Significance of electromagnetic potentials in quantum theory, Phys. Rev. 115,1959

[12] Jose,J. and Saletan, E., Classical Dynamics - A Contemporary Approach, Cambridge University Press,1998

[13] Lipson, S., Lipson, H. and Tannhauser, D., Optical Physics, Cambridge University Press, 1995

[14] http://en.wikipedia.org/wiki/Born-Oppenheimer_approximation (11/4/2013)