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arXiv:0909.2659v1 [hep-th] 14 Sep 2009

CP3-09-41

Variations on the Planar Landau Problem:

Canonical Transformations, A Purely Linear Potential

and the Half-Plane

Jan Govaertsa,b,1, M. Norbert Hounkonnouband Habatwa V. Mweenec

aCenter for Particle Physics and Phenomenology (CP3),

Institut de Physique Nucl´ eaire, Universit´ e catholique de Louvain (U.C.L.),

2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium

E-mail: Jan.Govaerts@uclouvain.be

bInternational Chair in Mathematical Physics and Applications (ICMPA–UNESCO Chair),

University of Abomey–Calavi, 072 B. P. 50, Cotonou, Republic of Benin

E-mail: hounkonnou@yahoo.fr, norbert.hounkonnou@cipma.uac.bj

cPhysics Department, University of Zambia

P. O. Box 32379, Lusaka, Zambia

E-mail: habatwamweene@yahoo.com, hmweene@unza.zm

Abstract

The ordinary Landau problem of a charged particle in a plane subjected to a perpendicular

homogeneous and static magnetic field is reconsidered from different points of view. The rˆ ole

of phase space canonical transformations and their relation to a choice of gauge in the solution

of the problem is addressed. The Landau problem is then extended to different contexts, in

particular the singular situation of a purely linear potential term being added as an interaction,

for which a complete purely algebraic solution is presented. This solution is then exploited to

solve this same singular Landau problem in the half-plane, with as motivation the potential

relevance of such a geometry for quantum Hall measurements in the presence of an electric

field or a gravitational quantum well.

1Fellow of the Stellenbosch Institute for Advanced Study (STIAS), 7600 Stellenbosch, South Africa.

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1Introduction

The classic textbook example[1] of the quantum Landau problem has remained a constant source

of fascination and inspiration[2], in fields apparently so diverse as two dimensional collective

quantum fermionic systems[3, 4], the search towards a fundamental unification of gravity with

the other quantum interactions, or noncommutative deformation quantisation of geometries[5, 6].

The same algebraic structures are also realised in M-theory in specific limits of some background

field configurations[7]. It is in view of the latter developments as well as the phenomenology

of the integer and fractional quantum Hall effects that in recent years the Landau problem has

become once again the focus of intense interest.

Yet, there remain somewhat intriguing issues open even for the simple original Landau

problem. Consider thus a charged particle of mass m moving in an Euclidean plane of coordi-

nates (x1,x2) and subjected to a static and homogeneous magnetic field perpendicular to that

plane, with a component B along the right-handed perpendicular direction which, without loss

of generality (by choosing the plane orientation appropriately), may be taken to be positive,

B > 0 (this factor, B, is also normalised so as to absorb the charge of the particle). Denoting

by (A1(x1,x2),A2(x1,x2)) the components of a vector potential from which the magnetic field

derives, ∂1A2− ∂2A1= B, it is well known that the dynamics of the system is specified through

the variational principle from the following Lagrange function,

L =1

2m?˙ x2

1+ ˙ x2

2

?+ ˙ x1A1(x1,x2) + ˙ x2A2(x1,x2), (1)

with as Hamiltonian function for the canonically conjugate pairs of phase space variables (xi,pi)

(i = 1,2),

1

2m

H =

?

p1− A1(x1,x2)

?2+

1

2m

?

p2− A2(x1,x2)

?2. (2)

The usual discussion[1] considers the Landau gauge for the vector potentiel,

ALandau

1

= −Bx2,ALandau

2

= 0, (3)

in which case one has,

H =

1

2mp2

2+1

2mω2

c

?

x2+1

Bp1

?2, (4)

with the cyclotron frequency ωc = B/m. For the quantised system, by introducing the Fock

algebra generators

a =

?mωc

2?

??

ˆ x2+1

Bˆ p1

?

+

i

mωcˆ p2

?

,a†=

?mωc

2?

??

ˆ x2+1

Bˆ p1

?

−

i

mωcˆ p2

?

,(5)

with,

?

ˆ x1, ˆ p1

?

= i?I,

?

a, ˆ p1

?

= 0,

?

a, ˆ x1

?

= −i

?

?

2BI,

?

a,a†?

= I, (6)

such thatˆH = ?ωc

|n,p1? (n = 0,1,2,...) with an energy ?ωc(n+1/2) which is degenerate in p1—the famous Landau

levels—, the latter real variable p1labelling the ˆ p1eigenstates.

However what is puzzling, perhaps, about this solution is the fact that because of the free

particle plane wave component of the configuration space wave function representation of these

states related to the p1eigenvalue, none of these states is normalisable,

?a†a + 1/2?, it is clear that the energy spectrum is spanned by Fock states

?n,p1|m,p′

1? = δnmδ(p1− p′

1), (7)

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while this basis of states is non countable and their wave functions are localised only in the x2

direction (through the Gaussian factor and Hermite polynomials in the (x2+p1/B) variable) but

not at all in the x1where they display complete delocalisation (note also that the (ˆ x1, ˆ p1) sector

does not commute with the Fock algebra, only the conjugate momentum operator, ˆ p1, does).

And yet the classical trajectories of such a particle are circles of which the radius is function of

the energy of the solution, the angular frequency is ωc, and the magnetic center is pinned at a

static position in the plane dependent on the initial conditions. Hence rather than the above

quantum states, one should expect there ought to exist another basis of the energy eigenstates

which describes normalisable and localised wave functions.

Indeed as is well known, in the circular or symmetric gauge,

Asym

1

= −1

2Bx2,Asym

2

= +1

2Bx1,(8)

once expanded, the Hamiltonian,

H =

1

2m

?

p1+1

2Bx2

?2

+

1

2m

?

p2−1

2Bx1

?2

, (9)

coincides with that of a two dimensional spherically symmetric harmonic oscillator of angular

frequency ωc/2 to which a term proportional to its angular-momentum is added. Working in a

complex parametrisation of the plane, it is clear1that the system is then diagonalised with a

countable energy eigenspectrum of Fock states, possessing the same energy spectrum as above,

but now represented by wave functions which are all localised and normalisable (and in fact

centered onto the point (x1,x2) = (0,0)).

At first sight, what distinguishes the above two gauge choices at the quantum level is a

redefinition of the wave functions of quantum states by a pure phase factor, eiχ(x1,x2), related to

the gauge transformation mapping the two choices of vector potentials into one another,

Asym

i

= ALandau

i

+ ∂iχ,χ(x1,x2) =1

2Bx1x2. (10)

The phase factor eiχbeing singular at the point at infinity in the plane, could be thought to be

the reason for the non normalisability and non localisability of energy eigenstates in the Landau

gauge. However, being a pure phase, such a phase redefinition alone cannot explain why out of

a localised and normalised wave function in the symmetric gauge one obtains a delocalised and

non normalisable one in the Landau gauge.

In Section 2 this question is addressed in detail, and resolved. Then in Section 3, using the

understanding gained from Section 2, and mostly to set notations for later use, the original Landau

problem is extended by adding an interaction potential energy which combines that of a spherically

symmetric harmonic well and a linear potential. When the harmonic well is removed an apparent

puzzle arises, the resolution of which is discussed in Section 4 using a purely algebraic approach.

To the best of our knowledge, the present algebraic solution—as opposed to a wave function

solution of the Schr¨ odinger equation in the Landau gauge—for the Landau problem extended

with a linear potential is not available in the literature. Finally, using the insight provided by

this construction and motivated by some physical considerations, Section 5 discusses the same

linearly extended Landau problem in the half-plane. The paper ends with some Conclusions.

1This is detailed in Section 2.

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2The Ordinary Landau Problem

2.1 A general choice of gauge

With the Lagrangian defined in (1), let us consider the following general class of gauge choices

for the vector potential,

A1(x1,x2) = −1

2B (x2− x2) + ∂1χ(x1,x2),A2(x1,x2) =1

2B (x1− x1) + ∂2χ(x1,x2). (11)

Here (x1,x2) are two constant parameters representing the position of a particular point in the

plane, about which configuration space wave functions representing the Fock states to be iden-

tified hereafter are centered and localised. Furthermore, χ(x1,x2) is an arbitrary real function

representing a possible gauge redefinition of the chosen vector potential. Note that the parame-

ters (x1,x2) could also be absorbed into that gauge transformation function, but it is useful to

keep these two constants explicit. Clearly the previous symmetric gauge corresponds to the values

(x1,x2) = (0,0) and χ = 0, while the Landau gauge to (x1,x2) = (0,0) and χ = −Bx1x2/2.

Incidentally, it may easily be checked that the Euler–Lagrange equations of motion that

derive from (1) are gauge invariant, namely independent both from (x1,x2) and χ(x1,x2), as it

should of course.

For what concerns the classical Hamiltonian formulation of the system, the Hamiltonian

reads,

H =

1

2m

?

p1+1

2B (x2− x2) − ∂1χ

?2

+

1

2m

?

p2−1

2B (x1− x1) − ∂2χ

?2

, (12)

where the phase space variables (xi,pi) possess canonical Poisson brackets, {xi,pj} = δij(i,j =

1,2). Introducing now the following new parametrisation of phase space,

ui= xi− xi,πi= pi− ∂iχ(xi), (13)

which defines yet again canonically conjugate pairs of variables,

?

ui,uj

?

= 0,

?

ui,πj

?

= δij,

?

πi,πj

?

= 0, (14)

one has,

H=

1

2m

1

2m

?

?π2

π1+1

2Bu2

?+1

?2

2mω2

+

1

2m

?

π2−1

2Bu1

?−1

?2

=

1+ π2

2

c

4

?u2

1+ u2

2

2ωc(u1π2− u2π1). (15)

The system has thereby been brought into the form it has in the symmetric gauge centered at

(x1,x2) = (0,0), independently of the original choice of gauge. Note well that this includes the

Landau gauge, however now with a choice of phase space canonical coordinates which differs from

that which led to (4). This point is addressed more specifically hereafter.

The resolution of the quantised system is now straightforward. Given the quantum com-

mutation relations,

?

one first introduces the cartesian Fock algebra generators,

ˆ ui, ˆ πj

?

= i?δijI,ˆ u†

i= ˆ ui,ˆ π†

i= ˆ πi,(16)

ai=1

2

?

B

?

?

ˆ ui+2i

Bˆ πi

?

,a†

i=1

2

?

B

?

?

ˆ ui−2i

Bˆ πi

?

,(17)

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such that

?

ai,a†

j

?

= δijI, (18)

while,

ˆH =1

2?ωc

?

a†

1a1+ a†

2a2+ 1

?

+1

2i?ωc

?

a†

1a2− a†

2a1

?

. (19)

Next one introduces the chiral Fock algebra generators,

a±=

1

√2(a1∓ ia2),a†

±=

1

√2

?

a†

1± ia†

2

?

, (20)

such that,

?

a±,a†

±

?

= I,

?

a±,a†

∓

?

= 0. (21)

A direct substitution2then finds,

ˆH = ?ωc

?

a†

−a−+1

2

?

.(22)

Consequently, given the orthonormalised Fock state basis |n−,n+? (n±= 0,1,2,...) defined by

1

?n−!n+!

these states diagonalise the energy eigenspectrum of the system,

|n−,n+? =

?

a†

−

?n−?

a†

+

?n+|0?,a±|0? = 0,?0|0? = 1, (23)

ˆH|n−,n+? = E(n−)|n−,n+?,E(n−) = ?ωc

?

n−+1

2

?

. (24)

Hence indeed the same energy spectrum as in the Landau gauge is obtained, however now

with a countable basis of eigenstates which are all normalisable and localised in the plane. More

specifically, it may be shown[8] that in the configuration space representation the wave functions

of these chiral Fock states are given as,

?x1,x2|n−,n+? =(−1)m

√2π?

?

m!

(m + |ℓ|)!u|ℓ|/2eiℓθe−1

2uL|ℓ|

m(u), (25)

where ℓ = n+−n−, m = min(n−,n+) = n−+(ℓ−|ℓ|)/2 and L|ℓ|

polynomials, while,

m(u) are the generalised Laguerre

u =mωc

2?

?

(x1− x1)2+ (x2− x2)2?

,eiθ=

(x1− x1) + i(x2− x2)

?

(x1− x1)2+ (x2− x2)2.(26)

Clearly all these states are thus indeed localised and centered at the point (x1,x2), and normal-

isable, independently of the chosen gauge for the vector potential, including the Landau gauge.

This result is achieved by having identified the appropriate canonical phase space transformation

which undoes any gauge transformation away from the symmetric gauge, while at the same time

moving the set of localised Fock states to be centered at any given point in the plane.

2The inverse relations expressing ˆ ui and ˆ πi in terms of (a±,a†

±) are easily worked out.

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2.2 The solution in the Landau gauge

In terms of the general parametrisation for a gauge choice in (11), the Landau gauge as defined

in the Introduction corresponds to the function

χ(x1,x2) = −1

2B (x1− x1)(x2+ x2). (27)

Consequently, one then finds,

π1

=p1+1

2B (x2+ x2),

p2+1

2B (x1− x1),

π1+1

2Bu2= π1+1

π2−1

2B (x2− x2) = p1+ Bx2,

π2

=

2Bu1= π2−1

2B (x1− x1) = p2, (28)

so that indeed,

H =

1

2m(p1+ Bx2)2+

1

2mp2

2. (29)

Given these relations in the Landau gauge, it is now possible to express the operators ˆ xi

and ˆ piin terms of the cartesian and chiral Fock operators introduced above. One then finds,

ˆ x1= x1+

?

?

B

?

a1+ a†

1

?

,ˆ x2= x2+

?

?

B

?

a2+ a†

2

?

, (30)

ˆ p1

=−1

−1

2i

√

?B

?

?

a1− a†

a2− a†

1

?

?

−1

−1

2

√

?B

?

?

a2+ a†

2

?

?

− Bx2,

ˆ p2

=

2i

√

?B

2

2

√

?Ba1+ a†

1

. (31)

We then have,

ˆ p2= −

?

?B

2

?

a−+ a†

−

?

,ˆ x2+1

Bˆ p1= −i

?

?

2B

?

a−− a†

−

?

,(32)

so that the (a,a†) Fock generators defined in (5) in the Landau gauge correspond to,

a = −ia−,a†= ia†

−. (33)

Hence we have indeed that

ˆH = ?ωc

?

a†

−a−+1

2

?

= ?ωc

?

a†a +1

2

?

, (34)

explaining why the same values for the energy spectrum are obtained in both constructions for

the quantum solution. However, in the discussion as recalled in the Introduction the degeneracy

of Landau levels is accounted for in terms of the eigenstates of ˆ p1, namely,

ˆ p1= −i

?

?B

2

?

a+− a†

+

?

− Bx2,(35)

rather than the Fock states |n+? of the Fock algebra (a+,a†

solution irrespective of the choice of gauge. Therefore when expressed in terms of these Fock

states, the solution to the eigenvalue equation,

+) as obtained in the previous general

ˆ p1|p1? = p1|p1?,p1∈ R, (36)

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involves an infinite linear combination of all Fock states |n+? which is not normalisable.

In other words, the reason why the usual solution to the Landau problem in the Landau

gauge leads to states which, within each of the Landau levels, are not normalisable nor localised,

is not at all related to that particular choice of gauge. Rather, it is because that choice of

gauge naturally leads one to use such a canonical parametrisation of phase space which upon its

canonical quantisation produces a basis of energy eigenstates which, in each Landau level, are

not normalisable nor localised. However this singular character in the choice of basis within each

Landau level is avoided by an appropriate canonical transformation which upon its canonical

quantisation produces a basis of energy eigenstates which, as Fock states, are all normalisable

and localised irrespective of the choice of gauge. It is thus coincidental that precisely in the

Landau gauge, the generic canonical phase space parametrisation valid for any choice of gauge is

just not manifest enough, so that one is lead rather onto a path towards another construction of

a solution for energy eigenstates which are no longer normalisable nor localised.

This analysis thus also shows that it is preferable to consider in all cases a parametrisation of

the general choice of gauge as in (11), which in effect is a gauge transformed form of the symmetric

gauge. One is then assured that if energy eigenstates are not normalisable or localised, there is

actually a physical justification or meaning to such a singular character, rather than being due

to some inappropriate choice of canonical parametrisation of phase space.

3 The Landau Problem with a Quadratic and Linear Potential

Given the above understanding of the preferred choice of gauge, let us now consider an extension

of the Landau problem which includes an interaction potential energy, V (x1,x2), still leading to

linear equations of motion, whether at the classical level or the quantum one in the Heisenberg

picture. In order to remain consistent with the rotational invariance of the original problem, this

potential consists of a spherically symmetric harmonic well of angular frequency ω0> 0 centered

at the origin (x1,x2) = (0,0), to which a linear term is added, lying—by an appropriate choice

of planar coordinates (x1,x2)—in the x2direction,

V (x1,x2) =1

2mω2

0

?x2

1+ x2

2

?+ γx2. (37)

Here γ is a real constant parameter, setting the strength of a constant pull onto the particle

in the (−x2) direction (for positive γ). This linear potential may correspond, for instance, to a

constant electric field lying inside the plane and along the x2direction. Another possibility is a

gravitational potential term if the plane is tilted with respect to the horizontal direction by some

angle α, in which case one has γ = mg cosα if x2increases up-wards, g > 0 being the gravitational

acceleration. These two examples thus indicate to which types of physical configurations such a

linear potential could correspond.

Choosing for the vector potential the symmetric gauge as in (8) does not lead to a straight-

forward resolution of either the Hamiltonian or the quantum dynamics. Indeed, since that choice

is centered onto the point (x1,x2) = (0,0), it clashes with the fact that because of the linear term

in the potential energy, the total potential energy—still spherically symmetric—is centered onto

a minimal position given by,

x1= 0,x2= −

γ

mω2

0

, (38)

since,

V (x1,x2) =1

2mω2

0

?

x2

1+

?

x2+

γ

mω2

0

?2?

−

γ2

2mω2

0

.(39)

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Obviously classical trajectories will then be centered onto that static average position in the

plane. Consequently, it is preferable to adapt the choice of symmetric gauge in the following way,

A1(x1,x2) = −1

2B

?

x2+

γ

mω2

0

?

,A2(x1,x2) = +1

2Bx1. (40)

The Hamiltonian is then of the form,

H=

1

2m

?

p1+1

2B

?

?+1

?

x2+

γ

mω2

0

??2

+

1

2m

?

p2−1

2Bx1

?2?

?2

+1

2mω2

0

?x2

1+ x2

2

?+ γx2

=

1

2m

?p2

?

1+ p2

2

2mω2

?

γ

x2

1+

?

?

x2+

γ

mω2

0

−1

2ωc

x1p2−x2+

mω2

0

?

p1

−

γ2

2mω2

0

, (41)

where ω =

?ω2

0+ ω2

c/4.

The diagonalisation of this quantum Hamiltonian is now straightforward enough3. Given

the Heisenberg algebra [ˆ xi, ˆ pj] = i?δijI, let us first introduce the following cartesian Fock algebra,

this time in terms of the effective angular frequency ω rather than the cyclotron one, ωc,

a1=

?mω

2?

?

γ

ˆ x1+

i

mωˆ p1

i

mωˆ p2

?

?

,a†

1=

?mω

?mω

2?

?

?

ˆ x1−

i

mωˆ p1

γ

mω2

?

,

a2=

?mω

2?

?

ˆ x2+

mω2

0

I +

,a†

2=

2?

ˆ x2+

0

I −

i

mωˆ p2

?

,(42)

which are such that,

?

ai,a†

j

?

= δijI. (43)

Introduce now once again the chiral or helicity Fock algebra operators

a±=

1

√2(a1∓ ia2),a†

±=

1

√2

?

a†

1± ia†

2

?

,(44)

?

a±,a†

±

?

= I. (45)

A simple substitution then finds for the quantum Hamiltonian,

ˆH = ?ω

?

a†

+a++ a†

−a−+ 1

?

−1

2?ωc

?

a†

+a+− a†

−a−

?

−

γ2

2mω2

0

, (46)

which is thus diagonalised on the basis of Fock states |n−,n+? (n±= 0,1,2,...) constructed out

of the chiral Fock algebra,

ˆH|n−,n+? = E(n−,n+)|n−,n+?,E(n−,n+) = ?ω+n−+ ?ω−n++ ?ω −

γ2

2mω2

0

, (47)

where,

ω+= ω −1

2ωc,ω−= ω +1

2ωc. (48)

3Had one not chosen the symmetric gauge centered onto the point in (38), there would have remained terms

linear in ˆ p1 inˆ H spoiling the simplicity of the present solution.

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The energy eigenspectrum thus consists of normalisable and localised states. As a matter

of fact, the configuration space wave functions, ?x1,x2|n−,n+?, of these chiral Fock states are

given as in (25), with this time the following definition for the two variables u and θ,

u =mω

?

?

x2

1+

?

x2+

γ

mω2

0

?2?

,eiθ=

x1+ i

?

?

x2+

γ

mω2

0

?

?

x2

1+x2+

γ

mω2

0

?2. (49)

It is also of interest to consider the time evolution of the quantum phase space operators

(ˆ xi, ˆ pi) in the Heisenberg picture. Given the above expression for the quantum Hamiltonian, the

time evolution of each of the Fock algebra operators is readily identified, leading to,

ˆ x1(t)=

1

2

?

?

mω

γ

mω2

?

a+e−iω−t+ a−e−iω+t+ a†

+eiω−t+ a†

−eiω+t?

+eiω−t+ a†

,

ˆ x2(t)=−

0

I +1

2i

?

?

mω

?

a+e−iω−t− a−e−iω+t− a†

−eiω+t?

,

,

ˆ p1(t)=−1

1

2mω

2imω

?

?

?

mω

?

a+e−iω−t+ a−e−iω+t− a†

+eiω−t− a†

−eiω+t?

−eiω+t?

ˆ p2(t)=

?

mω

?

a+e−iω−t− a−e−iω+t+ a†

+eiω−t− a†

.(50)

Of course, these expressions provide the explicit solutions to the linear Hamiltonian equations

of motion of the system, whether at the classical level, or the quantum level in the Heisenberg

picture. Note well that all the above operators (a±,a†

commutation relations specified either in the Schr¨ odinger picture, or the Heisenberg picture at

initial time t = 0.

All these expressions reproduce also those of the ordinary Landau problem of Section 2.1,

provided however the limits in ω2

0→ 0 and γ → 0 are taken appropriately. First the linear

potential term needs to be removed, γ → 0, and only then is the spherically symmetric well to

be flattened out, ω2

0→ 0. All the expressions above are then smoothly mapped back to those

of Section 2.1. In other words, by first bringing the equilibrium point of the total spherically

symmetric potential back to the origin of the plane, (x1,x2) = (0,0), namely by first removing

the linear contribution, and only then removing the spherical well, one reproduces the original

Landau problem.

±) are defined by the initial Heisenberg

However when the limits are considered in the reverse order, one immediately runs into

singularities. Indeed, note that when first the spherical well is flattened out while still keeping

the constant force acting on the particle, ω2

arise and the operators (a2,a†

±) are then no longer well defined. Nor is thus the

general solution in (50) and the chiral Fock states |n−,n+?.

Classically, by first removing the spherical well the particle is being set free—it is not

longer confined within the well—, and being subjected to a constant force inside the plane in

conjunction with the magnetic force which is always perpendicular to its velocity, the net result

is a circular motion around a magnetic center which rather than being static as in the ordinary

Landau problem, now moves at a constant velocity in a direction perpendicular to both the

magnetic field and the constant force, namely in our case along the x1direction with the velocity,

0→ 0 but γ ?= 0, singularities in the quantity γ/(mω2

2), hence (a±,a†

0)

˙ xc

1= −γ

B,

˙ xc

2= 0. (51)

8

Page 10

In terms of the above solution considered at the classical level, in the limit ω2

so that the actual classical solution acquires a linear time dependence—the one describing the

motion of its magnetic center at a constant velocity—combined with a periodic circular motion of

angular frequency ω+→ ωconce again, about that moving magnetic center. Applying a Galilei

boost taking the system to the inertial frame of the magnetic center, one recovers the ordinary

Landau problem. Note that the motion of the magnetic center is along the x1axis, but with a

value for x2which is a function of the initial conditions for the classical trajectory.

Rather than considering trying to apply to the above quantum solution, in particular the

construction of its chiral Fock states, a singular limiting procedure which at the classical level

produces out of the general solutions in (50) the correct ones when ω2

equations of motion are linear and thus identical whether for the classical or the quantum system

it is more straightforward to immediately consider the situation with ω2

classical level, and out of its solutions construct the appropriate realisation of the quantum system

in that singular case of the extended Landau problem. Such an approach is also simpler than

trying to apply to the ordinary Landau problem a Galilei boost from the magnetic center frame

back to the initial frame in which the dynamics of the system is being considered.

0→ 0 one has ω−→ 0,

0= 0, since the Hamiltonian

0= 0 and γ ?= 0 at the

4 The Landau Problem with a Linear Potential

In the symmetric gauge, the Lagrangian of the system now reads,

L =1

2m?˙ x2

1+ ˙ x2

2

?−1

2B (˙ x1x2− x1˙ x2) − γx2, (52)

hence the Hamiltonian,

H =

1

2m

?

p1+1

2Bx2

?2

+

1

2m

?

p2−1

2Bx1

?2

+ γx2. (53)

Trying to apply to the quantised version of the system the same types of operator redefinitions

as those of Section 3 runs into the difficulty that the term linear in γx2remains non diagonal in

whatever Fock state basis being considered. In order to tackle this issue, in the same way as was

discussed in Section 2.2 for the ordinary Landau problem in the Landau gauge, first a canonical

transformation of phase space parametrisation is required, which readily provides at the quantum

level the diagonalised Hamiltonian, hence the solution to the quantum dynamics of the system.

Rather than specifying this canonical transformation still at the classical level and in terms

of Poisson brackets, let us already define it at the quantum level for the quantum operators and

their commutation relations in the Schr¨ odinger picture, or the Heisenberg picture at time t = 0.

Consider then the following definitions, from the variables (ˆ xi, ˆ pi) to the variables (ˆ xc

1, ˆ xc

2,a−,a†

−),

ˆ xc

1

=

1

2ˆ x1+

1

2ˆ x2−

?mωc

?mωc

1

mωcˆ p2,

1

mωcˆ p1−

?1

?1

ˆ xc

2

=

γ

mω2

1

mωcˆ p2

1

mωcˆ p2

c

,

a−

=

2?

2ˆ x1−

?

?

+

i

mωc

i

mωc

?mωc

?mωc

2?

?

?

ˆ p1+1

2mωcˆ x2+γ

ˆ p1+1

ωcI

?

?

,

a†

−

=

2?

2ˆ x1−

−

2?

2mωcˆ x2+γ

ωcI

. (54)

9

Page 11

The inverse relations are,

ˆ x1

=ˆ xc

1+

?

?

?

2mωc

?

a−+ a†

−

?

,

ˆ x2

=ˆ xc

2− i

−γ

1

2mωcˆ xc

?

2mωc

?

a−− a†

−

?

,

ˆ p1

=

ωcI −1

2mωcˆ xc

2−1

?

2) and (a−,a†

2imωc

?

?

a−+ a†

?

2mωc

?

?

a−− a†

−

?

,

ˆ p2

=

1−1

2mωc

2mωc

?

−

.(55)

It then follows that the two sectors (ˆ xc

have,

1, ˆ xc

−) commute with one another, while we

?

ˆ xc

1, ˆ xc

2

?

= −i?

BI,

?

a−,a†

−

?

= I. (56)

Hence indeed this reparametrisation of phase space is a canonical transformation preserving

canonical Poisson brackets (one may rescale, say ˆ xc

Related to the magnetic center sector, (ˆ xc

2), we also have the following Fock algebra generators,

2, by the factor (−B = −mωc), if ones prefers).

1, ˆ xc

ˆ xc

1=

?

?

2mωc

?

a++ a†

+

?

,ˆ xc

2= i

?

?

2mωc

?

a+− a†

+

?

, (57)

which are such that,

?

a+,a†

+

?

= I. (58)

These operators (a+,a†

problem which in that context has no time dependence, but acquires one in the present case

because of the constant force of strength γ which indeed sets into motion the magnetic center.

Note that ˆ xc

1corresponds to the magnetic center position along the x1axis, while the contribution

ˆ xc

2to ˆ x2corresponds to its position along the x2axis. The contribution (−γ/ωc= −mγ/B) to ˆ p1

corresponds to the velocity momentum of the magnetic center, m˙ xc

to the left-handed chiral rotating mode with angular frequency ωcof the ordinary Landau problem

in the magnetic center inertial frame

A direct substitution of these relations in the Hamiltonian finds,

+) correspond to the right-handed chiral mode of the ordinary Landau

1. Finally, (a−,a†

−) correspond

ˆH = ?ωc

?

a†

−a−+1

2

?

+ γˆ xc

2+1

2m

?γ

B

?2. (59)

The physical meaning of each of these contributions should be clear enough. The very last term

corresponds to the kinetic energy of the magnetic center moving at constant velocity of norm

|γ|/B. The term before the last represents the potential energy along the x2direction, γx2, of

the magnetic center position along that axis. And finally the very first contribution with the two

terms in parentheses measures the excitation energy of the usual Landau levels of the ordinary

Landau problem, as seen from the magnetic center inertial frame.

This expression for the quantum Hamiltonian also makes it clear which basis of states

diagonalises that operator, hence solves the quantum dynamics of the system. Given the Fock

states |n−? associated to the (a−,a†

the magnetic center position along the x2axis, the basis of the space of quantum states which

diagonalises the dynamics is spanned by the states |n−,xc

normalisation,

?n−,xc

−) Fock algebra, and position eigenstates, ˆ xc

2|xc

2? = xc

2|xc

2?, for

2? (n−= 0,1,2,..., xc

2∈ R), with the

2|m−,x′c

2? = δn−,m−δ(xc

2− x′c

2). (60)

10

Page 12

One has,

ˆH|n,xc

2? = E(n,xc

2)|n,xc

2?,E(n,xc

2) = ?ωc

?

n +1

2

?

+ γxc

2+1

2m

?γ

B

?2. (61)

The fact that the magnetic center component of these quantum states is not normalisable

makes now perfect physical sense. Indeed, the motion of that magnetic center is that of a free

particle with a predetermined velocity set by the ratio (−γ/B) in the x1direction. Hence in the

configuration space representation states possess wave functions with a plane wave component

in that direction, which is not normalisable, and leads to the above δ function normalisation for

energy eigenstates. Having chosen from the outset not to work in the Landau gauge guarantees

without ambiguity that this lack of normalisability is indeed related to a physical feature of the

solution rather than a not totally appropriate choice of phase space parametrisation.

Given the Hamiltonian, it is also possible to determine the time dependence of the phase

space operators in the Heisenberg picture. One finds,

ˆ x1(t)=ˆ xc

1−

γ

BtI +

?

2mωc

I −1

?

?

2mωc

?

a−e−iωct+ a†

−eiωct?

,

,

ˆ x2(t)=ˆ xc

2− i

−γ

ωc

1

2mωcˆ xc

?

?

a−e−iωct− a†

−eiωct?

?

mωc

?

mωc

ˆ p1(t)=

2mωcˆ xc

2γtI −1

2−1

2imωc

?

?

a−e−iωct− +a†

a−e−iωct− a†

−eiωct?

−eiωct?

,

ˆ p2(t)=

1−1

2mωc

?

?

, (62)

namely,

ˆ xc

1(t) = ˆ xc

1−γ

BtI,ˆ xc

2(t) = ˆ xc

2. (63)

In the expressions for ˆ xi(t) one may recognise the solution to the ordinary Landau problem (the

terms involving a−and a†

−), valid in the magnetic center inertial frame, to which is added the

Galilei boost with the constant velocity of the magnetic center towards the inertial frame with

the potential energy γx2, and the initial position of that magnetic center along both the x1and

x2axes. Incidentally, and as was indicated at the end of the previous Section, this is in fact how

the change of variables (54) was identified initially. Writing out the classical solution for xi(t)

precisely in that way, and then identifying what are the ensuing expressions for pi(t) given that

p1(t) = m˙ x1(t) +1

2Bx2(t),p2(t) = m˙ x2(t) −1

2Bx1(t), (64)

all in a manner that meets all Hamiltonian equations of motion, the appropriate operators in the

Heisenberg picture are identified. Upon substitution into the quantum Hamiltonian, one then is

bound to find the quantum solution for it as well.

Note also that in the same spirit as that of the entire discussion so far, the above solution

of the singular Landau problem extended with a linear potential has remained purely algebraic,

without the need to solve the differential Schr¨ odinger wave equation. To the best of our knowledge,

this specific approach and construction for the extended singular Landau problem is not available

in the literature.

As a passing remark of some interest as well, note that given a projection onto any subspace

of Hilbert space corresponding to a Landau sector at fixed level n−, namely onto the subspace

spanned by the states |n−,xc

2? for a fixed n−and for all xc

2∈ R, in effect the only remaining degrees

11

Page 13

of freedom are those of the magnetic center, ˆ xc

ordinary Moyal–Voros plane of noncommutative geometry[5, 6], [ˆ xc

is readily established through the present discussion without the need of any actual calculation

of projected matrix elements, in contradistinction to the usual derivation of this result available

in the literature[9].

Finally, it now becomes feasible without much difficulty to explicitly work out the config-

uration space wave functions for all energy eigenstates, given the expressions of the operators

(ˆ xc

−) in terms of (ˆ xi, ˆ pi). This task would rather be a great deal more involved were

one to consider from the outset the differential Schr¨ odinger equation eigenvalue problem given

the Hamiltonian in the form of (53). Without going here into the details of the calculation, let us

only mention that in a first step one works out the wave function for the lowest Landau sector,

|n−= 0,xc

finds the wave functions for all states |n−,xc

the position eigenstates of the configuration space basis as is usual,

1and ˆ xc

2, which obey the commutation relation of the

1, ˆ xc

2] = −i(?/B)I. This result

1, ˆ xc

2,a−,a†

2?, as a function of xc

2. Applying then the operator a†

2?, including their normalisation. When normalising

−onto that solution, one readily

?x1,x2|x′

1,x′

2? = δ(x1− x′

1)δ(x2− x′

2),ˆ xi|x1,x2? = xi|x1,x2?,(65)

one finds,

?x1,x2|n−,xc

2?=

?mωc

×e

π?

?1/4?mωc

−imωc

?

x1(xc

2π?

?1/2 (−i)n

2x2+

mω2

√2nn!

×

2−1

γ

c)e−mωc

2?(x2−xc

2)2Hn

??mωc

?

(x2− xc

2)

?

, (66)

Hn(u) being the Hermite polynomial of order n.

Hence these energy eigenstates are localised only in the x2direction, while in the x1direction

they are totally delocalised and non normalisable, since they propagate in time in that direction

as a free particle of predetermined constant velocity (−γ/B). The probability density of these

states thus also looks like a series of (n+1) parallel stripes with exponentially smooth edges, and

invariant under translations along the x1axis.

Having constructed a complete solution of this singular Landau problem extended by a

linear potential, note how all these results are smooth in the parameter γ, and indeed reproduce

in the limit γ → 0 those of the ordinary Landau problem. Clearly in that limit the states |n−,xc

remain non normalisable, because the right-handed chiral sector (a+,a†

rather in terms of the operator ˆ xc

2, namely by having required the magnetic center position to be

sharp in the x2direction, hence totally delocalising its position along x1, since the two coordinates

of the magnetic center obey a Heisenberg algebra and do not commute. However in the limit

γ = 0 each of the Landau sectors distinguished by n−becomes once again energy degenerate,

allowing for another choice of energy eigenstate basis in the magnetic center sector. Choosing for

it once again the orthonormalised Fock state basis |n−,n+?, one recovers precisely exactly the

same solution as that constructed in Section 2.1 (with (x1,x2) = (0,0)). However the construction

of the present Section is useful even for the ordinary Landau problem, when a sharp rectilinear

edge is introduced in the plane, as we now discuss.

2?

+) has been diagonalised

5 The Landau Problem in the Half-Plane with a Linear Potential

A noteworthy feature of the energy spectrum (61) is that it is unbounded below, and yet the

quantum system (as well as the classical system) remains stable because the magnetic force

combines with the constant force of strength γ to keep the particle rotating periodically around

12

Page 14

a magnetic center that moves at constant velocity along the x1 axis. Clearly, the reason for

this unboundedness in the energy is that the particle may “fall off the plane” in one of the x2

directions, so to say, a way of putting this fact which is particularly appropriate in case the

constant force is indeed that of gravity.

A manner in which to avoid this unboundedness is to restrict the range of x2, namely

consider now the Landau problem on the half-plane with still the linear potential. Assuming now

that γ > 0, let us therefore restrict to the x2≥ X2half-plane for some value of X2∈ R, and

reconsider the solution of the quantised system. Such a situation is also of physical interest. Given

that the Landau problem is of relevance to the quantum Hall effect, in particular in its integer and

even fractional manifestations, combining the magnetic field with a constant force acting inside

the plane, be it electrical or gravitational, may allow for interesting properties of that collective

quantum fermion phenomenon to manifest themselves. In the gravitational context by tilting the

quantum Hall device towards the vertical direction, one is setting up a gravitational quantum well

in combination with the quantum Hall effect. Given that the energy quantisation of gravitational

quantum states in a gravitational well has been observed already with ultra-cold neutrons[10],

a quantum Hall set-up may provide an interesting alternative to such measurements, provided

the orders of magnitude for any effect are large enough to be observable. Of course, given the

weakness of gravity, an electric field stands a much better chance to display any such interesting

effects.

Clearly the change of variable specified in (54) is still in order in this case, leading to

the Hamiltonian in (59). However there is a subtlety now, given the boundary at x2 = X2.

Since one has to restrict now to the quantum Hilbert space of configuration space wave functions

that vanish at that boundary as well as inside the excluded domain, x2 < X2, the conjugate

momentum operator ˆ p2does not possess a self-adjoint extension, and is in fact only symmetric

on that space[11]. Indeed, as the differential operator (−i?∂2), the operator ˆ p2maps quantum

states out of that Hilbert space, while we have, for any two states |ψ? and |ϕ? represented by

functions ψ(x2) and ϕ(x2) in the (ˆ x2, ˆ p2) sector,

?ϕ|ˆ p2ψ?=

?+∞

X2

dx2ϕ∗(x2)

?

−i?

d

dx2ψ(x2)

?

=−i?

?+∞

?+∞

X2

d(ϕ∗(x2)ψ(x2)) +

?+∞

X2

dx2

?

−i?

d

dx2ϕ(x2)

?∗

ψ(x2)

=−i?

X2

d(ϕ∗(x2)ψ(x2)) + ?ˆ p2ϕ|ψ?. (67)

Given that both wave functions ψ(x2) and ϕ(x2) mush vanish at x2= X2and x2→ +∞ (the

latter condition applies since states must be normalisable at least in the x2direction), it follows

that ˆ p2is indeed a symmetric operator.

However among those operators contributing to the quantum Hamiltonian still given as in

(59), this ambiguity affects only the (a−,a†

−) operators, which are thus no longer adjoints of one

another, since each maps outside the considered space of quantum states. However, the other

operator involved in diagonalising the Hamiltonian, namely ˆ xc

self-adjointness in ˆ p2since it is given as

2, is not affected by that lack of

ˆ xc

2=1

2ˆ x2−

1

mωcˆ p1−

γ

mω2

c

, (68)

which is an expression that does not involve the operator ˆ p2, in contradistinction to the operators

(a−,a†

−). Consequently, one may still consider the space of eigenstates of ˆ xc

2, whose wave functions

13

Page 15

are given as in the construction of the previous Section, as a factor in the tensor product structure

providing a basis of energy eigenstates. For the remaining separable factor in that tensor product,

even though one may no longer exploit the existence of a Fock vacuum annihilated by a−in order

to diagonalise the Hamiltonian, it is rather the latter Hamiltonian which needs diagonalisation,

a procedure which does not necessarily require a construction of Fock states representing a Fock

algebra which in the present case does not exist. As will be seen hereafter, in contradistinction

to the (a−,a†

−) operators, the Hamiltonian operator itself is not affected by that issue and does

possess a self-adjoint extension[11].

Writing the quantum Hamiltonian in the form,

ˆH =1

2?ωc

?

a†

−a−+ a−a†

−

?

+ γˆ xc

2+1

2m

?γ

B

?2, (69)

and using the explicit expressions for a−and a†

formation to find,

−in (54), one undoes part of the canonical trans-

ˆH =1

2mω2

c

?1

2ˆ x1−

1

mωcˆ p2

?2

+

1

2m

?

ˆ p1+1

2mωcˆ x2+γ

ωc

?2

+ γˆ xc

2+1

2m

?γ

B

?2.(70)

Let us now consider the diagonalisation of this operator by working in the configuration space

wave function representation for quantum states, ψ(x1,x2). Since energy eigenstates are certainly

eigenstates of ˆ xc

2, their wave functions certainly separate as

ψE,xc

2(x1,x2) = e

−imωc

?

x1(xc

2−1

2x2+

γ

mω2

c)ϕE,xc

2(x2), (71)

E denoting their energy eigenvalue. A direct substitution in the stationary Schr¨ odinger equation

in the configuration space representation then reduces to, for any given value of xc

2,

?

−?

2m

d2

dx2

2

+1

2mω2

c(x2− xc

2)2+ γxc

2+1

2m

?γ

B

?2?

ϕE,xc

2(x2) = E ϕE,xc

2(x2), (72)

where one must also meet the condition ϕE,xc

for the energy values E. Note that indeed this operator possesses a self-adjoint extension for this

choice of boundary conditions[11].

Introducing the notations,

2(x2= X2) = 0 which will imply a quantisation rule

u =

?2mωc

?

(x2− xc

2),a = −

1

?ωc

?

E − γ xc

2−1

2m

?γ

B

?2?

,(73)

the above eigenvalue equation becomes

?d2

du2−

?1

4u2+ a

??

ϕE,xc

2(u) = 0.(74)

The general solution to this equation is a linear combination of the two parabolic cylinder functions

U(a,u) and V (a,u)[12]. However since wave functions are required to vanish at x2→ +∞, only

the U(a,u) branch is allowed4. Hence the solution is of the form,

ϕE,xc

2(x2) = N(E,xc

2)U(a,u),(75)

4Both functions U(a,u) and V (a,u) diverge as u → −∞, unless a = −n − 1/2 for n = 0,1,2,... in which case

only U(a,u) also vanishes in that limit, and in fact reduces[12] to U(−n − 1/2,u) = 2−n/2e−u2/4Hn(u/√2).

14

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