Quantum Oscillator on $DC P^n$ in a constant magnetic field

a r X i v :h e p -t h /0406184v 1 22 J u n 2004Quantum Oscillator on I CP n in a constant magnetic ?eld

Stefano Bellucci 1,Armen Nersessian 2,3and Armen Yeranyan 2

1

INFN-Laboratori Nazionali di Frascati,P.O.Box 13,I-00044,Frascati,Italy 2

Yerevan State University,Alex Manoogian St.,1,Yerevan,375025,Armenia 3Yerevan Physics Institute,Alikhanian Brothers St.,2,Yerevan,375036,Armenia

Abstract

We construct the quantum oscillator interacting with a constant magnetic ?eld on complex projective

spaces I CP N ,as well as on their non-compact counterparts,i. e.the N ?dimensional Lobachewski

spaces L N .We ?nd the spectrum of this system and the complete basis of wavefunctions.Surprisingly,

the inclusion of a magnetic ?eld does not yield any qualitative change in the energy spectrum.For

N >1the magnetic ?eld does not break the superintegrability of the system,whereas for N =1

it preserves the exact solvability of the system.We extend this results to the cones constructed

over I CP N and L N ,and perform the (Kustaanheimo-Stiefel)transformation of these systems to the

three-dimensional Coulomb-like systems.

Introduction The harmonic oscillator plays a fundamental role in quantum mechanics.On the other hand,there are few articles related with the oscillator on curved spaces.The most known generalization of the Euclidian oscillator is the oscillator on curved spaces with constant curvature (sphere and hyperboloid)[1]given by the potential V Higgs =ω2r 20x 20,?x 2+x 20=r 20,?=±1.(1)This system received much attention since its introduction (see for a review [2]and refs.therein)and is presently known under the name of “Higgs oscillator”.Recently the generalization of the oscillator to K¨a hler spaces has also been suggested,in terms of the potential [3]V osc =ω2g ˉa b ?ˉa K?b K.(2)Various properties of the systems with this potential were studied in Refs.[3,4,5,6].It was shown that on the complex projective spaces I CP N such a system inherits the whole set of rotational symmetries and a part of the hidden symmetries of the 2N ?dimensional ?at oscillator [3].In Ref.[4],the classical solutions of the system on I CP 2,L 2(the noncompact counterpart of I CP 2)and the related cones were presented,and the reduction to three dimensions was studied.Particulary,it was found that the oscillator on some cone related with I CP 2(L 2)results,after Hamiltonian reduction,in the Higgs oscillator on the three-dimensional sphere (two-sheet hyperboloid)in the presence of a Dirac monopole ?eld.In Ref.[5]we presented the exact quantum mechanical solutions for the oscillator on I CP 2,L 2and related cones.We also reduced these quantum systems to three dimensions and performed their (Kustaanheimo-Stie?el)

transformation to the three-dimensional Coulomb-like systems.The “K¨a hler oscillator”is a distinguished system with respect to supersymmetrisation as well.Its preliminary studies were presented in [6,3].

In this paper we present the exact solution of the quantum oscillator on arbitrary-dimensional I CP N ,L N and related cones in the presence of a constant magnetic ?eld .The study of such systems is not merely of academic interest.It is also relevant to the higher-dimensional quantum Hall e?ect.This theory has been formulated initially on the four-dimensional sphere [7]and further included,as a particular case,in the theory of the quantum Hall e?ect on complex projective spaces [8](see,also [9]).The latter theory is based on the quantum mechanics on I CP N in a constant magnetic ?eld.Our basic observation is that the inclusion of the constant magnetic ?eld does not break any existing hidden symmetries of the I CP N -oscillator and,consequently,its superintegrability and/or exact solvability are preserved.

To be more concrete,let us consider ?rst the (classical)oscillator on I R 2N =I C N .It is described by the symplectic structure

?0=dπa ∧dz a +d ˉπa ∧d ˉz a (3)

and the Hamiltonian

H =πˉπ+ω2z ˉz .

(4)

1

It has a symmetry group U(2n)given by the generators of SO(2n)rotations

J+ ab =ˉz aπb,J?

ˉaˉb

=z aˉπb,J aˉb=iz bπa?iˉπbˉz a(5)

and the hidden symmetries

I+

ab

=πaπb+ω2ˉz aˉz b,I?=ˉπaˉπb+ω2z a z b,(6)

I aˉb=πaˉπb+ω2ˉz a z b.(7) The oscillator on the complex projective space I CP N(for N>1)[3]and the one on Lobacewski space L N are de?ned by the same symplectic structure as above,see eq.(3),with the Hamiltonian

H=gˉa bˉπaπb+ω2r20r2

(1+?zˉz)(δab+?z aˉz b),?=±1.(8)

The choice?=1corresponds to I CP N,and?=?1is associated to L N.This system inherits only part of the rotational and hidden symmetries of the I C N-oscillator given,respectively,by the following constants of motion:

J aˉb=i(z bπa?ˉπbˉz a),I aˉb=2J+a J?b2ˉz a z b,(9)

where J+a=iπa+i?(ˉzˉπ)ˉz a,J?a=ˉJ+a are the translation generators.It is clear that J aˉb de?nes the U(N) rotations,while I aˉb is just a I CP N counterpart of(7).

In order to include a constant magnetic?eld,we have to leave the initial Hamiltonian unchanged,and replace the initial symplectic structure(3)by the following one:

?B=?+iBg aˉb dz a∧dˉz b,(10) where g aˉb is a K¨a hler metric of the con?guration space.It is easy to observe,that the inclusion of a constant magnetic?eld preserves only the symmetries of the I C N-oscillator generated by J aˉb and I aˉb.On the other hand,the inclusion of the magnetic?eld preserves all the symmetries for the oscillator on I CP N. Hence,in the presence of a magnetic?eld the oscillators on I C N and I CP N look much more similar,than in its absence1.Hence we could be sure that the I CP N-oscillator preserves its classical and quantum exact solvability in the presence of a constant magnetic?eld.

Below,we formulate the Hamiltonian and quantum-mechanical systems,describing the I CP N-and L N-oscillators in the constant magnetic?eld and present their wavefunctions and spectra.We also extend these results to the cones and discuss some related topics.

Oscillator in a constant magnetic?eld:I CP N and L N

The classical K¨a hler oscillator in a constant magnetic?eld is de?ned by the Hamiltonian

H=g aˉbπaˉπb+ω2gˉa b Kˉa K b,(11) and the Poisson brackets corresponding to the symplectic structure(10)

{πa,z b}=δb a,{ˉπa,ˉz b}=δb a,{πa,ˉπb}=iBg aˉb.(12) Its canonical quantization assumes the following choice of momenta operators:

πa=?i(ˉh?a+BK a/2), ˉπa=?i(ˉh?ˉa?BKˉa/2),(13)

where?a=?/?z a,?ˉa=?/?ˉz a,K a=?K/?z a,Kˉa=?K/?ˉz a.2The quantum Hamiltonian looks similar to the classical one

H=1

2?log(1+?zˉz),K a=

r20

1+?zˉz

,Kˉa=

r20

1+?zˉz

.(15)

The scalar curvature R is related with the parameter r20as follows:R=2?N(N+1)/r20.These systems possess the u(N)rotational symmetry generators

J aˉb=i?(z b πa?ˉz a ˉπb)+i?(z c πc?ˉz c ˉπc)δˉa b?Br201+?zˉz,(16) and the hidden symmetry de?ned by the generators

I aˉb= J+a J?b+ J?b J+a2ˉz a z b,(17) where J±a are the translations generators

J+a=i πa+i?ˉz a(ˉz ˉπ)?Br201+?zˉz, J?a=?i ˉπa?i?z a(z π)?Br201+?zˉz.(18) The Hamiltonian(14)could be rewritten as follows:

H=ˉh2x2N?1?(1+?x2)N?1??x2 J2+?(1+?x2)(2 J0+μB

2??

ˉh2μ2B

2ˉh

,2 J0=z a?a?ˉzˉa?ˉa,(20) J2is the quadratic Casimir of the SU(N)momentum operator for the N>1,and J= J0for N=1.In order to get the energy spectrum of the system,let us consider the spectral problem

HΨ=EΨ, J0Ψ=sΨ, J2Ψ=j(j+N?1)Ψ.(21)

It is convenient to pass to the2N?dimensional spherical coordinates(x,φi),where i=1,...,2N?1,x is a dimensionless radial coordinate taking values in the interval[0,∞)for?=+1,and in[0,1]for?=?1, andφi are appropriate angular coordinates.The convenient algorithm for the expansion of“Cartesian”coordinates to the spherical ones is described in Appendix.In the new coordinates the above system could be solved by the following choice of the wavefunction:

Ψ=ψ(x)D j s(φi),(22)

2?g

2

?g

where D j s(φi)is the eigenfunction of the operators J2, J0.It could be explicitly expressed via2N?dimensional Wigner functions,D j s(φi)= m i c m i D j{m?i},s(φi),where j,m i denote total and azimuth angular mo-menta,respectively,while s is the eigenvalue of the operator J0

m i,s=?j,?j+1,...,j?1,j j=0,1/2,1, (23)

Now,we make the substitution

x=tan[

√

√(sin[√?θ])1/2(24)

which yields the the following equation:

f′′+? ?E?j21?1/4?θ]?δ2?1/4?θ] f=0,(25) where

?E=2Er20

ˉh2+N2+μB2,δ2=

ω2r40

? 2,j1=2j+N?1.(26)

The regular wavefunctions,which form a complete orthonormal of the above Schroedinger equation,are of the form

ψ= C sin j1?1θcosδθ2F1(?n,n+δ+j1+1;j1+1;sin2θ),for?=1

C sinh j1?1θcosh?δ+2nθ2F1(?n,?n+δ,j1+1,tanh2θ),for?=?1,

(27) where n is the radial quantum number with the following range of de?nition:

n= 0,1,...,∞for?=1

0,1,...,n max=[(δ?j1?1)/2]for?=?1.(28) The normalization constants are de?ned by the expression

r2N0n!Γ2(j1+1)

2r20 (2n+2j+N+?δ)2?

ω2r40

ω2+

B20

spectrum)of that system,but makes it impossible to get the exact solution of its Schroedinger equation. So,opposite to the Higgs oscillator case,the K¨a hler oscillator on the two-dimensional sphere/hyperboloid behaves,with respect to the magnetic?eld,similarly to the planar one.

In free particle limit,i.e.forω=0,the energy spectrum is described by the principal quantum number J,which plays the role of the weight of the SU(N+1)group(when?=1),and the SU(N.1) group(when?=?1).For example,when?=1,the energy spectrum is of the form

E n+j,s=2

2r20

μ2B,J=n+j+|s+

μB

2?

log(1+?(zˉz)ν),ν>0;?=±1.(34)

The corresponding metric and oscillator potential are given by the expressions

g aˉb=νr20(zˉz)ν?1

zˉz(1+?(zˉz)ν)

ˉz a z b ,V osc=ω2r20

?θ]

?

,ψ=

f[θ]?N/2?1/4

?θ])N?1/2(cos[

√

?2ˉh2

+ 2s2?ˉh 2j21=(2j+N?1)2ν (N?1)2?4s2

2r20 (2n+j1+1)2+4s2r20δ(2n+j1+1)+ˉh B0s

ν?2(d y)2

ν)2,R0=r20.(39)

5

The Hamiltonian of the system is given by the expression

H MIC=1g πi√ν)2ν?γνν,(40) where

π=??ˉh?y3.(41) The coordinates of the initial and?nal systems are related as follows:

y= (zˉz)2(42)

The energy and coupling constantγof this system are de?ned by the energy and frequency of the respective four-dimensional oscillator.The quantum number s=0,±1/2,1,...becomes a?xed parameter (the“monopole number”),and instead of(23)one has

j=|s|,|s|+1,...;m=?j,?j+1,...,j?1,j.(43) It appears,that applying the KS-transformation to the four-dimensional oscillator in a constant magnetic ?eld,we have to get the modi?cation of the MIC-Kepler system on the three-dimensional hyperboloid (and related cones),which nevertheless remains superintegrable(exactly solvable).

Surprisingly,repeating the whole procedure,one can?nd that the inclusion of the magnetic?eld in the initial system yields,in the resulting system,a rede?nition of the coupling constantγand the energy

E only

γ=E osc

r20

(1?

s2

2ν

ˉh s,?2E=ω2+

?E osc

r40

(1+2

s2

r20ν

ˉh s.(44)

Using the expressions(27),one can convert the energy spectrum of the oscillator in the energy spectrum of the MIC-Kepler system

E=?2 γ??ˉh2(2n+j1+1)2/(4R0) 2

R0

+

ˉh2

Acknowledgments

We are indebted to Levon Mardoyan and Corneliu Sochichiu for useful conversations and their interest in this work.The work of S.B.was supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00131Quantum Spacetime,the INTAS-00-00254grant and the NATO Collaborative Linkage Grant PST.CLG.979389.The work of A.N.was supported by grant INTAS00-00262.

Appendix:The choice of the angular coordinates

It is convenient to pass to spherical coordinates by the use of the so-called“Smorodinsky’s trees method”. Let us illustrate this method on the simplest cases of N=2and N=3.Its extension to higher dimensions is straightforward.For choosing the appropriate angular coordinates we build Smorodinsky trees(?g.1).

Figure1:Smorodinsky trees for the cases N=2and N=3.

By means of these trees we express the Cartesian coordinates x1,...,x2N,via spherical ones,x,φi by the following rules.The ends of the top of the branches mark Cartesian coordinates.The stock corresponds to the radial coordinate x.Each node marks some angleφi.The angles marked by the top nodes have the range of de?nition[0,2π),the remaining angles have the range of de?nition[0,π/2). For the expansion of Cartesian coordinates in spherical ones we have to go to each top starting from the stock.When we are passing through a node(denoting theφangle)to the left,we must write cosφ, whereas when passing through a node to the right,we must write sinφ.

Explicitly,one has

for N=2

z1=x1+ix2,z2=x3+ix4;(46)

x1=x cos[β/2]cos[(α+γ)/2],x2=x cos[β/2]sin[(α+γ)/2],

x3=x sin[β/2]cos[(α?γ)/2],x4=x sin[β/2]sin[(α?γ)/2].

β∈[0,π),α∈[0,4π),γ∈[0,2π)

for N=3

z1=x1+ix2,z2=x3+ix4,z3=x5+ix6;(47)

x1=x cos[θ/2]cos[β/2]cos[(α+γ)/2],x2=x cos[θ/2]cos[β/2]sin[(α+γ)/2],

x3=x cos[θ/2]sin[β/2]cos[(α?γ)/2],x4=x cos[θ/2]sin[β/2]sin[(α?γ)/2].

x5=x sin[θ/2]cos[λ/2],x6=x sin[θ/2]sin[λ/2].

β,θ∈[0,π),α,λ∈[0,4π),γ∈[0,2π)

7

In these coordinates the operators J an J0look as follows: for N=2

J2=?

1

?β sinβ?sin2β

?2?γ2?2cosβ?2?α

,

for N=3

J2=?1

sinβ

?

?β +1?α2+?2?α?γ

?(49)

?1?λ2?1?θ

cos3[θ/2]sin[θ/2]?

?α+

?

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