Conﬁned one-dimensional harmonic oscillator as a two-mode ... Publications... · set of states...

Confined one-dimensional harmonic oscillator as a two-mode systemV. G. Gueorguieva�

Lawrence Livermore National Laboratory, Livermore, California 94550

A. R. P. Rau and J. P. DraayerDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803

�Received 10 June 2005; accepted 6 January 2006�

The one-dimensional harmonic oscillator in a box is possibly the simplest example of a two-modesystem. This system has two exactly solvable limits, the harmonic oscillator and a particle in a�one-dimensional� box. Each of the limits has a characteristic spectral structure describing the twodifferent excitation modes of the system. Near these limits perturbation theory can be used to findan accurate description of the eigenstates. Away from the limits it is necessary to do a matrixdiagonalization because the basis-state mixing that occurs is typically large. An alternative toformulating the problem in terms of one or the other basis set is to use an “oblique” basis that usesboth sets. We study this alternative for the example system and then discuss the applicability of thisapproach for more complex systems, such as the study of complex nuclei where oblique-basiscalculations have been successful. © 2006 American Association of Physics Teachers.

�DOI: 10.1119/1.2173270�

I. INTRODUCTION

The understanding of a quantum system is closely linkedto how well we can determine its eigenstates. Typically a setof basis states that works well in one limit fails in another.More general methods, such as variational schemes, pertur-bation theory, or fixed-basis matrix diagonalizations typicallybegin with a reasonable Hamiltonian and some appropriateset of basis states near one limit where they provide a gooddescription of the system. However, there are many ex-amples where the Hamiltonian has more than a single exactlysolvable limit. How to approach such examples is the topicof this paper.

A simple example is the hydrogen atom in an externalmagnetic field: With increasing field strength, particularly formagnetic fields exceeding a critical value of 2.35�105 T,the system changes from the spherical symmetry of the Cou-lomb problem to the cylindrical symmetry of the diamag-netic Hamiltonian.1 An example that occurs in a variety ofcondensed matter applications is a particle confined to twodimensions in an external magnetic field.2 In nuclear physicsthe interacting boson model classifies many nuclei accordingto one of three dynamical symmetries.3 What should be doneif the system is nowhere near any of the exact limits? Inthese situations, the problem may be approached by usingstates associated with all the appropriate nearby limits. Thisset of states will form an oblique �mixed-mode� basis for thecalculation.4,5 In general, such a basis is nonorthogonal andmay even be over complete. Nevertheless, as recent studies4,5

demonstrate, such oblique bases have merit. In this paper weconsider a pedagogically simple problem to illustrate the ob-lique basis approach.

The one-dimensional harmonic oscillator in a one-dimensional box has been used to illustrate different aspectsof mixing. Barton, Bray, and Mackane used the model tostudy the effects of distant boundaries on the energy levels ofa one-dimensional system.6 Studies have also been done forthe cylindrically symmetric system of a three-dimensionalharmonic oscillator between two impenetrable walls.7 How-ever, these studies did not discuss the bimodal nature of theproblem. Some authors have generalized the one-
dimensional harmonic oscillator problem by introducing
394 Am. J. Phys. 74 �5�, May 2006 http://aapt.org/ajp

time-dependent parameters in the Hamiltonian8 to go be-tween the two limiting cases of a free particle and the har-monic oscillator. The infinite square well and the harmonicoscillator have been considered as two limiting cases of apower-law potential within the context of wave packet col-lapses and revivals.9,10 Recent studies of modified uncer-tainty relations has also shown related and interestingbehavior.11

In this paper we demonstrate the oblique basis approachby considering the simple two-mode system of the one-dimensional harmonic oscillator in a box. In Sec. II we dis-cuss the concept as well as the exactly solvable limits of thistoy model. A qualitative discussion of the expected spectrumof the one-dimensional harmonic oscillator in a box is givenin Sec. III, along with an example spectrum and quantitativeestimates. In Sec. IV some specific problems related to thestructure of the Hilbert space are addressed. Section V con-tains specific toy model calculations and results as well as adiscussion of interesting behavior, similar to that observed innuclear structure.4,5 The discussion in Sec. V is focused on aquasiperturbative behavior and a coherent structure withinthe “strong mixing” region12 where the system is far from anexact limit. Our conclusions are discussed in Sec. VI andsuggested student problems are given in Sec. VII.

II. HARMONIC OSCILLATORIN A ONE-DIMENSIONAL BOX

We start with an abstract two-mode system. For simplicity,we assume that the Hamiltonian has two exactly solvablelimits,

H = �1 − ��H0 + �H1. �1�

Equation �1� is set up so that H→H0 in the limit �→0 andH →H1 when �→1. Near these two limits we can use stan-dard perturbation theory for one Hamiltonian perturbed bythe other.13 Somewhere between these two limits there usu-ally is a critical value of �=�c that is related to the stronglymixed regime of the system. Around the critical value thetwo limits of the system are equally important and perturba-
tion theory about one set of eigenstates is not applicable. In
394© 2006 American Association of Physics Teachers

this region there is strong mixing of the properties and statesof the two limits.12 The value of �c can be anywhere in theinterval �0,1�, a convenient choice being �c�1/2. Some-times, a further symmetry breaking Hamiltonian H2 can beexplicitly introduced by adding ��1−��H2 to H.

In Eq. �1� the variable � has been introduced to simplifythe discussion. In general, there will be more than one suchparameter in the Hamiltonian.14 Often the exactly solvablelimits are described as hypersurfaces in parameter space. Itcould even be that there are three or more exactly solvablelimits. For example, the interacting boson model3 for nuclearspectra has three exactly solvable limits.15,16 Another ex-ample with three exactly solvable limits is the commonlyused schematic interaction with nondegenerate single-particle energies, pairing two-body interactions,17 andquadrupole-quadrupole two-body interactions.18

Here we consider what is perhaps the simplest two-modesystem that shares the essential features of such problemswhile remaining pedagogically instructive. The Hamiltonianof a one-dimensional harmonic oscillator in a one-dimensional box of length 2L has the form19

H =1

2mp2 + VL�q� +

m�2

2q2, �2�

where VL�q� is a confining potential with the value zero for�q � �L and � for �q � �L, and � is the oscillator frequency.This system has two exactly solvable limits.

One limit of the toy model in Eq. �2� is �=0 for which itreduces to a free particle in a one-dimensional box of length2L,

H0 =1

2mp2 + VL�q� . �3�

The eigenvectors and energies are labeled by n=0,1 , . . .and are given by

�n�q� = ��1

Lcos�n + 1�

�q

2L n even,

�1

Lsin�n + 1�

�q

2L n odd,

�4�

En1D =

1

2m��n + 1�

�

2�2�

L�2

. �5�

This limit corresponds to extreme nuclear matter when theshort range nuclear force can be described as an effectiveinteraction represented by a square-well potential.20 We canthink of this limit as the one-dimensional equivalent of athree-dimensional model where the nucleons are confinedwithin a finite volume of space representing the nucleus. Re-cently such effective potentials for the Bohr Hamiltonianhave been used to introduce the X�5� and E�5� symmetries inthe critical point of quantum phase transitions.21

The other exactly solvable limit of Eq. �2� is the harmonicoscillator in one dimension when L→�,

H1 =1

2mp2 +

m�2

2q2. �6�

In dimensionless coordinates,

395 Am. J. Phys., Vol. 74, No. 5, May 2006

q → q̃�

m�, p → p̃�m � , �7�

we have

H1 = �12 �p̃2 + q̃2� . �8�

The eigenvectors and energies are labeled by n=0,1 , . . . andare given by

n�q� =� 1

bn ! 2n��Hn�q

b�exp�−

1

2

q2

b2� , �9�

EnHO = ��n +

1

2�, b =�

m�, �10�

where Hn are the Hermite polynomials. This limit is essen-tially the harmonic oscillator model for nuclei. Even in moresophisticated nuclear models,22,23 as far as single shell stud-ies are concerned, it is appropriate to consider the one-dimensional harmonic oscillator as representative of theSU�3� shell model.

III. SPECTRAL STRUCTURE AT DIFFERENTENERGY SCALES

Often the spectrum of a system has different characteris-tics over different energy regimes. This difference usuallyreflects the existence of different excitation modes of thesystem. For the toy model Hamiltonian in Eq. �2� we candefine three spectral regions:

�i� The spectrum of a particle in a one-dimensional boxas in Eq. �5� with a quadratic dependence on n, En

1D

n2.�ii� The spectrum of the one-dimensional harmonic oscil-

lator as in Eq. �10� with a linear dependence on n,En

HO n.�iii� The intermediate spectrum which is neither of the pre-

vious two types.

As shown in Fig. 1, we expect to see the particle in a boxspectrum at high energies. These energies correspond to thebox boundaries dominating over the harmonic oscillator po-tential. In this regime we can use standard perturbationtheory to calculate the energy for a particle in a box per-

Fig. 1. Two-mode toy system consisting of a particle in a one-dimensionalbox subject to a central harmonic oscillator restoring force �m=1�.

turbed by a harmonic oscillator potential. It can be shown

395Gueorguiev, Rau, and Draayer

that perturbation theory gives better results for higher energylevels. For n→� the first correction �En

1 approaches a con-stant value,

�En1 =

1

6m�2L2�1 −

6

�1 + n�2�2� →1

6m�2L2. �11�

Because En+10 −En

0� �n �V �n�, we estimate that perturbationcalculations are valid when

n �2m2�2L4

32�2 . �12�

This analysis is confirmed by the numerical calculationsshown in Fig. 2 where the perturbed particle in a box spec-trum provides a good description for n 6 for the case of2L /�=1 and �=4. �In this calculation and all that follow wechoose units such that m= =1.� Actually, the agreementextends to lower n as is often the case; perturbation theoryfrequently yields valid results well past its expected region ofvalidity. Note that the first order corrections are already closeto the limiting constant value of 1

6m�2L2. On the other hand,first-order perturbation clearly fails for the ground state. In-deed, an earlier study19 of the inadequacy of fixed basis cal-culations showed that adequate convergence for the groundstate when � is large requires a large number of basis statesof the one-dimensional box. This requirement is related tothe fact that for large values of �, or equivalently large val-ues of L, a large number of particle-in-a-box wave functions�sin and cos functions� are needed to obtain the correct ex-ponential fall off of the low-energy harmonic oscillator wave

Fig. 2. Exact energies of a two-mode system with 2L /�=1 and �=4 com-pared to the spectrum of the one-dimensional harmonic oscillator �left�,spectrum of the free particle in a one-dimensional box �right�, and the spec-trum as calculated within a first order perturbation theory of a free particle ina one-dimensional box perturbed by a one-dimensional harmonic oscillatorpotential. The lowest three eigenenergies of the two-mode system nearlycoincide with the one-dimensional harmonic oscillator eigenenergies, whilehigher energy states are better described as perturbations of the other limit ofa free particle in a one-dimensional box.

functions in the classically forbidden zone.


The intermediate spectrum would be observed when theharmonic oscillator turning points coincide with the walls ofthe box. Therefore, the critical energy that separates the twoextreme spectral structures is given by

Ec =m�2

2L2. �13�

Note that the constant energy shift m�2L2 /6 �see Eq. �11�� inthe energy of the high energy levels n�1 is one-third of thecritical energy �Ec /3�.

At low energies where the classical turning points of theoscillator lie far from the boundaries, we expect the spectrumto coincide with that of the oscillator as shown in Fig. 2. Thenumber of such nearly harmonic oscillator states is easilyestimated using Ec En

HO, which implies that

nmaxHO =

1

2

m�L2

−

1

2. �14�

There is also a compatible number of levels, usually largerthan nmax

HO , below the Ec corresponding to a free particle in abox. That is, Ec En

1D implies

nmax1D =

2

�

m�L2

− 1. �15�

However, these states are mixed by the harmonic oscillatorpotential.

If we use the ratio of the ground state energies, E0HO/E0

1D

=4m�L2 / ��2�, together with Eqs. �14� and �15�, the fol-lowing spectral situations apply:

�i� For �� /2�2�m�L2 /, there are levels below Ec cor-responding to the harmonic oscillator and the particlein a box such that E0

HO E01D. The oscillator levels

dominate the low energy spectrum.�ii� For � /2�m�L2 / � �� /2�2, there are only the

ground states E01D and E0

HO below Ec and E01D E0

HO.�iii� For 1�m�L2 / �� /2, there is only the ground state

of the harmonic oscillator E0HO below Ec.

�iv� For 0�m�L2 / �1, perturbation theory of a particlein a box should be applicable for all levels.

The dimensionless parameter �=m�L2 / thus plays a rolesimilar to the parameter � in Eq. �1�. In Eq. �1� the two limitsof H are at �=0 and �=1 with strong mixing at �=1/2. Inthe toy model we are considering, the two limits are �=0and �=� with strong mixing when 1�� /2�2. In thisrespect we can relate � to � using an expression of the form�=� / ��+�c�, where �c is the value of � in the strong mix-ing region. For example, we may choose �c=� /2.

A numerical illustration of the two-mode spectra with L=� /2 and �=4 is shown in Fig. 2. With these parameters,Eq. �14� gives nmax

HO =4.53. Thus we should see no more thanfour equidistant levels corresponding to a harmonic oscillatorspectrum.

With respect to the critical energy Ec, there is a moreexplicit classification of the spectral structure:

�i� Perturbed particle in a one-dimensional box spectrumfor energies E�Ec such that Eq. �12� holds.

�ii� One-dimensional harmonic oscillator spectrum in Eq.�10� for energies Ec�E such that Eq. �14� holds.

�iii� Intermediate spectrum for energies E�Ec.


IV. HILBERT SPACE OF THE BASIS WAVEFUNCTIONS

Before discussing Eq. �2� using an oblique basis, it is in-structive to discuss the harmonic oscillator problem in Eq.�6� using the wave functions for a free particle in a one-dimensional box as in Eq. �4�; and vice versa, solving theproblem of a free particle in a one-dimensional box in Eq.�3� using the wave functions for a particle in the harmonicoscillator potential given in Eq. �9�.

Due to the different domains of the wave functions, thereare some specific problems that need to be addressed. Forexample, using wave functions for a free particle in a one-dimensional box to solve the pure harmonic oscillator prob-lem may not be appropriate especially for high energy states,E�Ec. The reason is that all such functions vanish outsidethe box �see Fig. 3� unlike the oscillator wave functions,especially as the energy increases. The converse is also prob-lematic because oscillator functions that are nonzero outsidethe box will lead to infinite energy.

The influence of the boundary conditions on the propertiesof a quantum-mechanical system has long been recognized.It is well known that some problems with seemingly sepa-rable Hamiltonians may recouple due to the boundaryconditions.24

Some recent studies on confined one-dimensional systemsused truncation factors �a cutoff function� to enforce theboundary conditions and derived equations satisfied by thesefactors.6,25 Other authors used variational procedures to ob-tain the truncation factors26,27 or derived asymptotic esti-mates for multiparticle systems using the Kirkwood-Buckingham variational method.28 Somewhat differentapproaches focus on shape-invariant potentials and use su-persymmetric partner potentials to derive energy shifts andapproximate wave functions,29 as well as dependence of theground-state energy on sample size.30 In the following wediscuss the structure of the relevant Hilbert spaces when con-finement is present.

A. Harmonic oscillator in the one-dimensional box basis

Consider the harmonic oscillator in Eq. �6� using the wavefunctions for a free particle in a one-dimensional box in Eq.�4�. There are no practical difficulties for energies E�Ec as

Fig. 3. The harmonic oscillator wave functions spread outside the harmonicoscillator potential into the classically forbidden region; the particle in a boxwave functions are zero at and outside the box boundary.

defined in Eq. �13� as long as the turning points of the oscil-


lator are sufficiently deep into the box �sufficiently far fromthe walls of the box�. For E�Ec the basis wave functions arelocalized only on the interval �−L ,L� and thus cannot pro-vide the necessary spread over the width of the potential�Fig. 3�. This situation would be appropriate for the toymodel in Eq. �2�, but not for the pure harmonic oscillatorproblem in Eq. �6�.

One simple solution to the spreading problem is to con-tinue the basis wave functions using periodicity. In this waythe necessary spread of the basis wave functions can beachieved and the new basis will stay orthogonal but must berenormalized. If we continue the wave functions to infinity,normalization will require Dirac delta functions but for con-tinuation on a finite interval, the functions can be normalizedto unity as usual. However, these basis wave functions do notdecay to zero in the classically forbidden zone, which meansthat a significant number of basis wave functions are neededto account for the appropriate behavior within the classicallyforbidden zone.

An alternative is to change the support domain corre-sponding to nonzero values of the function by stretching orsqueezing, accomplished by a scaling of the argument of thebasis wave functions, x→x�n /L. In this way the supportbecomes �−L ,L�→ �−�n ,�n�, where �n is a scale factor forthe nth basis wave function in Eq. �4� and is estimated eitherfrom the width of the harmonic oscillator potential or byvariational minimization. Either way, the new set of basisfunctions will be nonorthogonal. In general, the set may noteven be linearly independent. However, for the basis func-tions discussed here, a linear dependence may not appear dueto the different number of nodes �zeros� for each wave func-tion. The number of nodes is not changed under the rescalingprocedure. Although the potential width scaling is simpler,its applicability is more limited than the variationally deter-mined one, which can be extended to more generalsituations.5

B. Particle in a box in the harmonic oscillator basis

When the choice of the basis is not made with appropriatecare, an operator that should be Hermitian may become non-Hermitian. Although this is unlikely for a finite shell-modelcalculation in nuclear physics using an occupation numberrepresentation,31 it is a problem when we wish to use a hardcore potential and a harmonic oscillator basis.15

Suppose that we want to solve the problem of a free par-ticle in a one-dimensional box �−L ,L� as given in Eq. �3�using the harmonic oscillator wave functions in Eq. �9�. Wefirst change the inner product of the wave functions,

�f ,g� = �−�

�

f*�x�g�x�dx → �−L

L

f*�x�g�x�dx . �16�

It is clear that the set of previously orthonormal harmonicoscillator wave functions n�q� in Eq. �9� will lose theirorthonormality and even their linear independence. The setof functions n�q� with support domain restricted to�−L ,L� and denoted by n�q ; �−L ,L�� become linearly de-pendent if L is so small that there is more than one functionn�q ; �−L ,L�� with the same number of nodes within�−L ,L�. Although this linear dependence can be handled, the
real problem is the loss of Hermiticity of the physically sig-

a�.

nificant operators. This non-Hermiticity is due to the behav-ior of the basis states at the boundary, mainly the nonvanish-ing of the wave functions at ±L.

To understand the loss of Hermiticity, we look at the off-diagonal matrix elements of the momentum operator p̂=−i �� /�q�. After some manipulations we have

�m, p̂n� = �p̂m,n� − i �m* �q�n�q��−L

L . �17�

It is clear from Eq. �17� that Hermiticity will be maintainedonly when all of the basis functions are zero at the boundaryof the interval �−L ,L�. �Actually, the necessary condition isthat the wave functions have the same value at ±L; they needto be zero only for an infinite potential wall.�

In our simple example all operators are built from themomentum operator p̂ and position operator q̂. Thus to en-sure Hermiticity, it is sufficient to make sure that p̂ is Her-mitian, which requires the basis wave functions to vanish atthe boundaries ±L. For this purpose we can look at the nodesof each basis wave function and scale it so that its outernodes are at the boundary points. From the nodal structure ofthe harmonic oscillator wave functions, it is clear that thefirst two wave functions �0 and 1� cannot be used becausethey have fewer than two nodes. Because the requirement

Fig. 4. Harmonic oscillator trial wave functions in blue �dark gray� adjusnodally adjusted �first three are phase shifted�, and �c� boundary adjusted usiin red �light gray� for a particle in a box are zero at ±1, as clearly seen in �

that the wave functions must be zero at the boundary is the


cornerstone of quantizing a particle in a one-dimensional boxas in Eq. �4�, it is not surprising that the nodally adjustedharmonic oscillator wave functions are very close to the ex-act wave functions for the free particle in a one-dimensionalbox as shown in Fig. 4.

Calculating the nodes of a function may become verycomplicated. To avoid problems with finding the roots, wecan evaluate the wave function at the boundary points, thenshift the wave function by a constant to obtain zeros at theboundary, �q�→�q�−�L�. This idea works well foreven parity wave functions, but must be generalized for oddparity by adding a linear term, �q�→�q�− ��L� /L�q.Thus, for a general function we can have �q�→�q�− �1+q /L��L� /2− �1−q /L��−L� /2. In Fig. 4 we show someof the resulting wave functions. Note that this proceduregives a new wave function that is well behaved in theinterval �−L ,L� and grows linearly with q outside the interval�−L ,L�. This behavior contrasts with the behavior of the cut-off function f�q� obtained in Ref. 6 where the function f�q�has an L /q singularity at the origin �q=0�. The use of acutoff function to enforce boundary conditions has been

a� according to the potential width En1D=�n

2L2 /2Þ�n= � /L2��1+2n�, �b��q�→�q�−�L��1+q /L� /2−�−L��1−q /L� /2. The exact wave functions

ted �ng

studied in Refs. 6 and 25 and provides an interesting integral


equation for the cutoff function. On the other hand, a simplecutoff function supplemented by a variational method seemsto be very effective.26–28

An alternative and more involved construction that relieson the Lanczos algorithm can be used. This algorithm isiterative and uses the Hamiltonian to generate a new basisstate from the previous state at each iteration. By using theboundary matching process that we have discussed, we cansuccessfully modify the usual Lanczos algorithm32 to solvefor the few lowest eigenvectors of the free particle in a one-dimensional box through an arbitrary, but reasonable, choiceof the initial wave function.5 The major modification is toproject every new function n+1=Hn into the appropriateHilbert space and subtract the components along any previ-ous basis vectors. After that, we can evaluate matrix ele-ments of H with these new basis vectors which clearly liewithin the correct Hilbert space. In this way we must doublethe number of scalar product operations compared to theusual Lanczos algorithm where the matrix elements of H arecalculated along with the complete reorthogonalization of thebasis vectors.

C. Oblique basis for the two-mode system

We have seen that a basis that is appropriate for one modeof our model system, a harmonic oscillator in a box, is notappropriate for the other mode. If we desire a description inthe critical mixed region characterized by Ec, a combinationof the two basis sets seems appropriate. This basis is referredto as an oblique basis, stemming from the geometrical anal-ogy. A two-dimensional space is usually described in termsof the Cartesian �x ,y� axes or in terms of any other orthogo-nal pair obtained by rotating these axes. Although orthogonalchoices are more convenient and familiar, any two axes, aslong as they are not linearly dependent, will suffice to de-scribe the space. Such a choice constitutes an oblique pair.When we mix both harmonic oscillator and particle in a boxstates, we have an oblique basis.

To use an oblique basis there are two main problems to beaddressed in order to have a proper Hilbert space. First, wemust ensure that any set of states that are derived from har-monic oscillator functions satisfies the boundary conditionsof the problem. For the particle in a box states the boundaryconditions are satisfied by construction. We have already dis-cussed a few ways to construct states with the correct bound-ary conditions. An interesting additional method would be touse Hermite functions with a noninteger index. For suchfunctions with index between n and n+1, the position of theouter node is between the outer nodes of the nth and �n+1�thHermite functions. The other problem is that after the chosenset of functions has been modified to satisfy the boundaryconditions, the orthonormality of these functions will mostlikely be destroyed. Even if the two basis sets were ortho-normal by themselves, they would not be orthonormal as awhole and may even be linearly dependent. Although thisproblem might seem serious, it has a well known solutionthrough the reorthonormalization of the basis or by proceed-ing with a generalized eigenvalue problem �see Eq. �24��.4

Our oblique basis consists of modified harmonic oscillator�MHO� basis states that satisfy the boundary conditionsalong with the basis states of a free particle in a box. To
obtain the wave functions and eigenenergies we solve the

generalized eigenvalue problem within this oblique basis.Schematically, these basis vectors and their overlap matrixcan be represented as

E = � e�,box basis

ei,MHO basis� �basis vectors� , �18�

� = � 1 �

�+ �� overlap matrix� , �19�

��i = e� · ei, �20�

�ij = ei · ej , �21�

H = �H�� H�j

Hi� Hij�

= � Hbox�box Hbox�MHO

HMHO�box HMHO�MHO� �Hamiltonian� ,

�22�

where �=1, . . . , dim�box basis� and i=1, . . . , dim�MHO ba-sis�.

In this notation the eigenvalue problem,

Hv = Ev �23�

with v=v�e�+viei takes the form

�H�� H�j

Hi� Hij��v�

v j � = E� 1 �

�+ ��v�

v j � �24�

which is a generalized eigenvalue problem.

V. DISCUSSION OF THE TOY MODELCALCULATIONS AND RESULTS

Despite the simplicity of the toy model in Eq. �2�, theharmonic oscillator in a box exhibits some of the essentialcharacteristics of a more complex system. Our interest lies inpart in problems associated with the use of fixed-basis cal-culations. One such problem is the slow convergence of thecalculations.19 If we can implement exact arithmetic, weneed not worry much about the slow convergence whenenough time, storage, and other computer resources are pro-vided. However, a numerical calculation that convergesslowly may be compromised by the accumulated numericalerror. Having the correct space of functions is of the utmostimportance in any calculation.

Having considered the main problems we might face instudying the simple toy model in Eq. �2�, we continue ourdiscussion with the spectrum for 2L /�=1 and �=4. We per-form our calculations using an increasing basis size until weachieve convergence of the eigenenergy of the desired lowenergy states. As we see in Fig. 2 the first three energy levelsare equidistant and almost coincide with the harmonic oscil-lator levels as expected from Eq. �14�. For these states thewave functions are also essentially the harmonic oscillatorwave functions. The intermediate spectrum is almost miss-ing. Above Ec, the spectrum is that of a free particle in aone-dimensional box perturbed by the harmonic oscillatorpotential. We find that an oblique-basis calculation repro-duces the lowest eight energy levels using 14 basis functions,seven nodally adjusted harmonic oscillator states, and seven
states of a free particle in a box. In contrast, a fixed-basis

calculation, using only the wave functions of a free particlein a one-dimensional box requires 18 basis states.

Due to the simplicity of the toy model, it does not appearas if the oblique-basis calculation has a big numerical advan-tage over calculations using the fixed basis of the box wavefunctions. There are two main reasons for why there is no bignumerical advantage: first there is a critical energy Ec thatseparates the two modes, and second the spectrum above Echas a regular monotonically increasing level spacing struc-ture.

The regular structure above Ec results in a very favorablesituation for the usual fixed-basis calculations because thedimension of the space needed to obtain the nth eigenvaluegrows as n+�. The parameter � is relatively small and doesnot depend strongly on energy. For example, the �=16 cal-culations need only �=15 extra basis vectors when calculat-ing any of the eigenvectors up to n=100. The relatively con-stant value of � can be understood by considering theharmonic oscillator potential as an interaction that createsexcitations out of the nth unperturbed box state. In such anexcitation picture, � is the number of box states with ener-gies in the interval En

0 and En0+�2 /2��n �x2 ��n�, where En

0 isthe nth unperturbed box state energy. There is a rapid decou-pling of the higher energy states from any finite excitationprocess that starts out of the nth state. This decoupling is dueto the increasing energy spacing of the box spectrum, whichresults in a limited number of states mixed by the harmonicoscillator potential. For the upper limit Ec /3 on �En

1, weestimate ��1/�3�nmax

1D .The sharp separation of the two modes �the presence of a

critical energy Ec� allows for the safe use of the harmonicoscillator states without any rescaling. Using unmodified har-monic oscillator states is especially acceptable when � isvery large because then the low energy states are naturallylocalized within the box. Therefore, instead of diagonalizingthe Hamiltonian in the box basis, we can just use the har-monic oscillator wave functions.

Figure 5 shows the absolute deviation �E=Enexact

−Enestimate of the calculated energy spectrum for �=16 and

estimate

Fig. 5. Absolute deviations of variously calculated energies from the exactenergy eigenvalues for �=16 and L=� /2 as a function of eigenstate numbern. The blue circles represent the deviation of the exact energy eigenvaluefrom the corresponding harmonic oscillator eigenvalue, �E=En

exact−EnHO, the

red diamonds are the corresponding deviation from the energy spectrum of aparticle in a box, �E=En

exact−En1D, and the green squares are the first-order

perturbation theory calculations.

L=� /2. Here, En represents one of the three energy


estimates: the harmonic oscillator EnHO, the particle in a one-

dimensional box En1D, and the first-order perturbation theory

result with the harmonic oscillator potential as a perturba-tion, En

1D+�2 /2��n �x2 ��n�. There are about 19 states thatmatch the harmonic oscillator spectrum, which is consistentwith the expected value from Eq. �14�. After n=20, pertur-bation theory gives increasingly better results for the energyeigenvalues. Figure 6 shows the relative deviation1−En

estimate /Enexact.

As expected, perturbation theory is valid for high energystates determined by Eq. �12�. For the high energy spectrum,the harmonic oscillator potential acts as a small perturbation.Thus the first-order corrections in the energy and the wavefunction are small. Figure 7 shows that the main componentof the 105th exact wave function comes from the 105th boxwave function, as it should in a region of small perturbations.

For low energy states, perturbation theory around the boxstates is not appropriate, and the harmonic oscillator statesare closer to the true states in this region. For 2L /�=1 and�=16, the first ten states are essentially the harmonic oscil-lator states to a very high accuracy. The next ten states stillhave high overlap with the corresponding harmonic oscilla-tor wave functions. For example, starting from 0.999 999 at

Fig. 6. As in Fig. 5 but for relative deviations from the exact energy eigen-values of the three calculations.

Fig. 7. Nonzero components of the 105th exact eigenvector in the basis of afree particle in a one-dimensional box. The parameters are �=16 and L
=� /2.

the tenth state, the overlap decreases to 0.880 755 at thetwentieth state. After that the overlap becomes small veryquickly. Figure 8 shows the structure of the third exact ei-genvector when expanded in the box basis. Notice that thethird box wave function is almost missing. An explanationlies in the structure of the harmonic oscillator functions,which are essentially exact in this region. Upon projectingthese functions in Eq. �9� onto the box functions in Eq. �4�,the results can be obtained in closed analytical form. Theintegrand consists of three factors, an even power of q, aGaussian, and a cosine. Although the oscillations of the Her-mite polynomial are of varying amplitude, the cosine hasevenly spaced nodes and antinodes of equal amplitude. As aresult, cancellations can take place between successive termsin the integrand. For the third oscillator function �the secondof even parity�, its one pair of nodes is reflected in the dipseen at n=3 in Fig. 8. Higher oscillator functions with morepairs of nodes can display correspondingly more dips in theprojected squared amplitudes.

This pattern of a small projection of the exact wave func-tion along the corresponding box wave function persists intothe transition region. This persistence may seem unexpectedconsidering that the first-order estimates of the energy levelsare relatively good. Note that in our two examples with �=4 �Fig. 2� and �=16 �Fig. 5�, the first-order corrections inthe transition region are already close to the limiting value ofm�2L2 /6, even though the corresponding box wave func-tions are not present in the exact wave function as shown inFig. 9.

From the results in Figs. 5 and 6, it seems that the transi-tion region is absent because first-order perturbation theorybecomes valid immediately after the breakdown of the har-monic oscillator spectrum. That first-order perturbationtheory gives good estimates for the energy levels in this tran-sition region is due to a coherent behavior. What is happen-ing in this region is a coherent mixing of box states by theharmonic oscillator potential.5,33,34 This coherent mixing isillustrated in Fig. 9 where we see that the probability distri-butions for a few consecutive states are very similar. In thissense we say that the corresponding particle in a box statesare coherently mixed. A more precise and detailed discussionof coherent structure behavior and quasistructures is in

Fig. 8. Nonzero components of the third harmonic oscillator �even parity�eigenvector expanded in the basis of a free particle in a one-dimensionalbox. The parameters are �=16 and L=� /2.

Ref. 5.


VI. CONCLUSION

We have studied the simplest two-mode system consistingof a one-dimensional harmonic oscillator in a one-dimensional box. Depending on the parameters of the twoexactly solvable limits, various spectral structures are ob-served. There is clear coherent mixing in the transition re-gion. It is remarkable that such a simple system can exhibitcoherent behavior similar to that observed in nuclei. Anoblique-basis set which includes both oscillator and particlein a box states allows us to use the correct wave functions inthe relevant low and high energy regimes relative to Ec andoffers a clear numerical advantage. The oblique-basismethod can be extended beyond two or more orthonormalbasis sets. Specifically, we can consider a variationally im-proved basis set by starting with an initially guessed basisstates. In the occupation number representation, which is of-ten used in the nuclear shell model,31 this variationally im-proved basis method seems inapplicable. But the method is

Fig. 9. The coherent structure with respect to the nonzero components of the25th, 27th, and 29th exact eigenvector in the basis of a free particle in aone-dimensional box. The parameters are �=16 and L=� /2.

of general interest because of its possible relevance to mul-


tishell ab initio nuclear physics, atomic physics, and generalquantum mechanical calculations. The oblique-basis methodmay also be related to renormalization-type techniques.5

VII. STUDENT PROBLEMS

�1� Section II views the two Hamiltonians in Eqs. �3� and �6�as limits of Eq. �2� for suitable choices of parameters.Express this feature in the form of Eq. �1� with the suit-able definition and choice of � and the appropriate modi-fication of � in Eq. �6�.

�2� Treat the harmonic oscillator potential as a perturbationand work out the first-order correction to the energies ofparticle in a box states. Hence, verify Eq. �12�.

�3� Project the third harmonic oscillator wave function �sec-ond even parity state with two nodes� onto the wavefunctions of a particle in a box to verify the structureshown in Fig. 8. The relevant integrals have products ofpowers of q, a Gaussian, and a cosine in q; the integra-tion limits may be taken to be ±�.

�4� As a variational counterpart to the oblique basis conceptdiscussed in this paper, the product of the ground statefunctions of a particle in a box and an oscillator may bechosen as a trial wave function. Use such a form to cal-culate the ground state energy of the system using theRayleigh-Ritz variational principle.

ACKNOWLEDGMENTS

This work was supported by the National Science Foun-dation under Grants Nos. PHY 0140300 and PHY 0243473,and the Southeastern Universities Research Association. Oneof the authors �A.R.P.R� thanks the Research School ofPhysical Sciences and Engineering at the Australian NationalUniversity for its hospitality during the writing of this paper.This work was partially performed under the auspices of theU. S. Department of Energy by the University of California,Lawrence Livermore National Laboratory under ContractNo. W-7405-Eng-48.

a�On leave of absence from Institute of Nuclear Research and NuclearEnergy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria.

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