Scaling behavior of the momentum distribution of a quantum Coulomb system in a confining potential
SScaling behavior of the momentum distribution of a quantum Coulomb system in aconfining potential
J. A. E. Bonart a,b , W. H. Appelt a , D. Vollhardt a , L. Chioncel a,c a Theoretical Physics III, Center for Electronic Correlations and Magnetism,Institute of Physics, University of Augsburg, 86135 Augsburg, Germany b BMW AG, 80807 Munich, Germany and c Augsburg Center for Innovative Technologies, University of Augsburg, 86135 Augsburg, Germany (Dated: May 12, 2020)We calculate the single-particle momentum distribution of a quantum many-particle system inthe presence of the Coulomb interaction and a confining potential. The region of intermediatemomenta, where the confining potential dominates, marks a crossover from a Gaussian distributionvalid at low momenta to a power-law behavior valid at high momenta. We show that for all momentathe momentum distribution can be parametrized by a q -Gaussian distribution whose parameters arespecified by the confining potential. Furthermore, we find that the functional form of the probabilityof transitions between the confined ground state and the n th excited state is invariant under scalingof the ratio Q /ν n , where Q is the transferred momentum and ν n is the corresponding excitationenergy. Using the scaling variable Q /ν n the maxima of the transition probabilities can also beexpressed in terms of a q -Gaussian. I. INTRODUCTION
The single-particle momentum distribution plays animportant role in our understanding of the ground-state properties of quantum many-particle systems [1,2]. It is defined as the average number of particleswith momentum k in an N-particle system, n ( k ) = (cid:104) Ψ | (cid:80) σ a † k σ a k σ | Ψ (cid:105) . Here the normalized N -particle stateof the system is represented by | Ψ (cid:105) and a † k σ ( a k σ ) arethe creation (annihilation) operators for particles withmomentum k and spin projection σ . In real space theone-particle density matrix ρ ( x , x (cid:48) ) = (cid:90) N (cid:89) i =2 d x i ψ † ( x , x , ..., x N ) ψ ( x (cid:48) , x , ..., x N )(1)measures the change of the N -particle wave functionwhen a particle is moved from x (cid:48) to x while allother particles are fixed. In homogeneous systems thistwo-point function depends only upon the separation: ρ ( x , x (cid:48) ) = ρ ( | x − x (cid:48) | ). Accordingly, the momentumdistribution and the one-particle density matrix are re-lated by Fourier transformation: n ( k ) = (cid:90) d x (cid:90) d x (cid:48) e i k · ( x − x (cid:48) ) ρ ( x − x (cid:48) ) . (2)The momentum distribution, eq. (2), is determined bya product of two field operators whose short-distance be-havior can be calculated exactly using renormalizationgroup methods [3–6]. The latter techniques when ap-plied in nuclear physics decouple the low- from the high-momentum degrees of freedom and leave the scatteringcross section invariant [2, 7–11]. Furthermore, the nu-clear momentum distributions calculated within the im-pulse approximation [12, 13] provide universal scalinglaws for the high-momentum tails [2] of one- and two-particle momentum distributions [4, 14]. These tails are the consequence of short range-correlations in the nu-clear wave functions [15]. Renormalization group argu-ments [16, 17] have also shown that high-momentum tailsof momentum distributions factorize into a product be-tween a universal function of momentum, which is de-termined by two-particle physics and a factor depend-ing on the low-momentum structure of the many-bodystate [6]. This observation goes back to Kimball [18, 19]who pointed out that when two particles are sufficientlyclose their interaction dominates, and the two-particleSchr¨odinger equation provides a reasonable starting pointto compute quantum mechanical observables from theknowledge of the pair-wave function.Experimental measurements of n ( k ) involve inelasticscattering processes with energy and momentum trans-fers larger than the characteristic length scale of the scat-terer. They determine the double differential scatteringcross section d σ/d Ω dω for a given infinitesimal solid an-gle d Ω and energy dω of the scattered particle, respec-tively. The incident energy and the scattering angle arefixed during the experiment, and the scattering cross sec-tion is measured as a function of the transferred mo-mentum and energy. The data analysis of the measuredscattering cross section generally employs the impulse ap-proximation [12, 13], which assumes that a single particleis struck by the scattering probe, and that the particlerecoils freely from the collision. Within the impulse ap-proximation the scattering cross section is proportionalto the Compton profile d σ/d Ω dω ∝ J ( k z ). The lattercan be calculated directly by integrating the momentumdistribution n ( k ) in a plane perpendicular to the scat-tering vector k z : J ( k z ) = (cid:82) (cid:82) n ( k ) dk x dk y . The propor-tionality implies that, whenever the measured scatteringcross section is modelled within the impulse approxima-tion [1, 2] and is found to be invariant under some scalingtransformation, the Compton profile will show the samescaling behavior.In this paper we investigate whether the momen- a r X i v : . [ c ond - m a t . o t h e r] M a y tum distribution of a Coulomb system, which yields theCompton profile by integration, also shows scaling be-havior. In high energy physics it is well known that thescaling of the scattering cross section is a consequenceof confinement (“Bjorken scaling”). Indeed, by assum-ing the existence of a simple confining potential for twopoint particles, Elitzur and Susskind [20] derived thescaling behavior of the resonance excitations found ex-perimentally in deep inelastic reactions [8, 21]. Mak-ing use of Kimball’s observation [18, 19] we will there-fore compute the momentum distribution of two inter-acting electrons by numerically solving the two-particleSchr¨odinger equation for a repulsive Coulomb interactionin the presence of a confining potential. In the follow-ing we work in atomic units, where the unit length is a = 1 Bohr(0 . × − m), the unit of mass is theelectron mass m , and the unit of energy is 1 Hartree(1 Ha = 27 . V = α | x/a | η . The condition η > α = 1 Ha. We will show that the momentum distribu-tion of a quantum many-particle system interacting viathe Coulomb interaction in the presence of a confiningpotential can be parametrized by a q -Gaussian distribu-tion whose parameters are determined by the confiningpotential.In Sec. II we compute the high-momentum tails of themomentum distribution, eq. (2), in the groundstate andshow that they obey scaling relations. A crossover in themomentum density from an ordinary Gaussian distribu-tion at small momenta to a power-law behavior at largermomenta occurs when the Coulomb potential dominatesthe confinement. In the cross-over region of size ≈ /a we find that the shape of the momentum distribution isdescribed by a q -Gaussian with k -dependent parameters.At large momenta we recover the exact results obtainedby renormalization group methods [4–6, 16, 17]. Usingthe solutions of the two-particle Schr¨odinger equation ofSec. II we show in Sec. III that when the confinementdominates the Coulomb interaction, the transition ma-trix elements into excited states ( n th -bound level) dueto momentum absorption also obey q -Gaussian distribu-tions, and we connect the q -parameter to the shape of theconfining potential. The q -Gaussian used to parametrizethe momentum distribution is characterized by parame-ters which are different from those used to fit the tran-sition probabilities between the bound states. Finally,in Sec. IV we relate these results to the recent observa-tion [22] that the Compton profiles of all alkali elementscan be collapsed onto a single curve which is describedby a q -Gaussian. II. GROUND STATE: KIMBALL’S APPROACHTO THE MOMENTUM DISTRIBUTION
We consider non-relativistic electrons with Coulombinteraction whose Hamiltonian reads [18, 19]: H = − ¯ h m N (cid:88) i =1 ∇ i − N (cid:88) i =1 N I (cid:88) I =1 e Z I | x i − R I | + (cid:88) i 2, the Schr¨odinger equationbecomes (cid:18) − ∂ ∂x + V ( x ) + 1 x (cid:19) ψ n ( x ) = E n ψ n ( x ) . (4) FIG. 1. Probability density | ψ n ( x ) | of the pair wave func-tions for n = 0 , , V ( x ) together with the repul-sive Coulomb term 1 /x , which is singular at x = 0. Thecorresponding energies E n of the ground state and the firstexcited states are shown on the right. For a better visibilitythe probability densities are separated along the vertical axis. The solutions to eq. (4) are denoted by ψ n ( x ) with thecorresponding eigenenergy E n , are shown in Fig. 1. Theansatz for the total wave function introduced by Kim-ball [18, 19] separates the dependence on the relative co-ordinates from that of the center of mass motionΨ n ≡ Ψ n ( x , x , x , ..., x N )= Ψ n ( x , X , x , ..., x N ) (cid:39) Φ( X , x , ..., x N ) ψ n ( x ) . (5)The one-particle density matrix for the relative coor-dinates is then given by ρ ( x , x (cid:48) ) (cid:39) (cid:90) d X N (cid:89) i =3 d x i Φ † ( X , x , ..., x N ) ψ † n ( x )Φ( X , x , ..., x N ) ψ n ( x (cid:48) )= ρ ψ † n ( x ) ψ n ( x (cid:48) ) , (6)where ρ represents the integral of the N -particle wavefunction over all coordinates x i and X . From ρ ( x , x (cid:48) ) weobtain the two-particle correlation function g ( x ), whichis defined as g ( x ) = ρ ( x , x ) = ρ ψ † n ( x ) ψ n ( x ) and is pro-portional to the probability of two particles being at adistance x . The momentum distribution is computed ac-cording to eq. 2.Fig. 2 (a) shows the momentum dependence of n ( k )for the confining potential V ( x ) = α | x / a | η with η = 2,where k is the magnitude of k . One clearly sees a Gaus-sian regime at small k , which is followed by a crossoverinto the asymptotic region at large k . In order to specifythe momentum dependence by a unique functional formwe fit the momentum distribution with a q -Gaussian [23].To determine the ( q, β ) parameters we collect k valuesinto “bins” that are characterized by an average value¯ k . The latter quantity is computed from the intervalin which the fitting to the q -Gaussian form is performedand which contains at least five points in the low momen-tum and an order of magnitude more (fifty) points in theasymptotic region. The initial fitting parameters of the i th bin are the final parameters of the ( i − th ¯ k -bin. Fora given bin the same values q (¯ k ) , β (¯ k ) parametrise themomentum dependence as: [24] n q (¯ k ) ,β (¯ k ) ( k ) = 1 √ β (¯ k ) C q (¯ k ) exp q (¯ k ) (cid:0) − β (¯ k ) k (cid:1) . (7)For arbitrary values of q , the q -exponential is definedas exp q ( x ) = [1 + (1 − q ) x ] / (1 − q ) . In eq. 7 C q (¯ k ) is anormalization constant and β (¯ k ) controls the width ofthe distributionIn Fig. 2 (b) we plot the momentum dependence of the( q, β )-parameters. We see that for low momenta q (¯ k ) = 1while β (¯ k ) = 0 . 49. In fact, the exp q -function becomesthe exponential function in the limit of q → 1, wherebythe Gaussian distribution is recovered. The low momen-tum region corresponds to large distances. In this casethe Coulomb interaction is negligible and the solutionsbecome plane waves [18, 19], which form Gaussian wavepackages leading to a Gaussian momentum dependence. FIG. 2. (a) Computed momentum distribution n ( k ) of elec-trons in the presence of a confining potential ( α = 1Ha, η = 2). The momentum distribution is fitted by a q -Gaussianin the entire momentum region. (b) Dependence of the q, β -parameters on the average momentum ¯ k . The low momentumregion shows an ordinary Gaussian behavior ( q = 1). In theintermediate region ( k min ≈ . /a < k < k max ≈ . /a ) q -Gaussian type of fits are possible with momentum dependentparameters. At large momenta the power-law dependence isrecovered. In the crossover region, i.e., in the range ¯ k min ≈ . /a to ¯ k max ≈ . /a , a smooth transition between Gaussianand power law behavior is observed in the momentum de-pendence of the ( q, β ) parameters.For large values of k > k max the q -Gaussian distribu-tion has a power law dependence f q,β ( x ) | x →∞ ∝ x / (1 − q ) ,which in our case amounts to constant values of the pa-rameters q (¯ k ) = 1 . 50 and β (¯ k ) = 0 . 06. Thus, at largemomenta the power-law behavior is recovered [18, 19],and the asymptotic behavior of the momentum densityagrees with the result obtained by the renormalizationgroup approach [16, 17]. We note that the q -Gaussianfits and the corresponding q (¯ k ) , β (¯ k )-parameters differ fordifferent values of the confining potentials (values of theparameter η ). Nonetheless, constant values of q (¯ k ) , β (¯ k )are obtained for the low momentum region as well as theasymptotic region.In the following section we will show that the transi-tion matrix elements between the ground state and the n th -energy level caused by absorption of a momentum,calculated from the solutions of eq. 4, also obey scaling.This result was previously found by Elitzur and Susskindwithin a simplified parton model [20]. Here we computethe full transition probability and demonstrate that thescaling functions for the maxima of the transition proba-bilities can also be expressed by q -Gaussians. Since tran-sition probabilities are connected to the scattering crosssection from which the Compton profiles follow, our cal-culation proves that the Compton profile also scales, pro-vided that the potential energy due to the confinementdominates the Coulomb interaction. III. EXCITED STATES: ELITZUR-SUSSKINDBOUND STATE RESONANCES Elitzur and Susskind employed a simple confining po-tential [20] to explain the scaling of the resonance excita-tions in deep inelastic reactions [8, 21]. In their simplifiedmodel the probability for transitions between the boundstates in the confining potential was computed using thedipol approximation and was shown to be compatiblewith the scaling of resonant excitations [20, 25, 26] indeep inelastic reactions. Contrary to Ref. [20] in whichthe WKB approximation was used, we solve eq. 4 numer-ically for a pair of point particles with masses m , m ina potential which is sufficiently deep to bind states. Weassume that the momentum Q is absorbed by one of theparticles, and the bound pair is lifted to the n th -level, atan energy ν n = E n − E . Using the solutions of eq. 4 weevaluate the matrix elements F (cid:0) ν n , Q (cid:1) for the transi-tion into the n th bound level due to the absorption of amomentum Q . We note that the solutions of eq. 4 pro-duce bound states [27] irrespective of the sign of the con-fining potential. The transition probability is the squareof the transition matrix element: T (cid:0) ν n , Q (cid:1) = | (cid:104) ψ n | e i Q · x | ψ (cid:105) | = | (cid:104) ψ n | e i Q x | ψ (cid:105) | . (8)Here ψ and ψ n describe the ground state and the n th bound state of the confining potential, respectively, with Q i = Q m i / ( (cid:80) i m i ). For electrons m = m whereby Q i = Q / Q thetransition probability T ( ν n , Q ) leads to n discrete reso- nances. The numerical results are presented in Fig. 3. FIG. 3. Transition probability T ( ν n , Q ) computed for theconfining potential with η = 2 in the presence of the Coulombinteraction. The scaling direction is seen by projection intothe ( ν n , Q ) plane. In Ref. [20] it was noted that the matching betweenthe phase of ψ n and the exponential factor e i Q x impliesa linear relation (“scaling direction”) between the squareof the transferred momentum, Q , and the excitation en-ergy ν n . The scaling direction therefore represents a linein the ( ν n , Q )-plane which we illustrate in the upper partof Fig. 3. For any other direction in the ( ν n , Q )-planethe transition probability decays exponentially (no phasematching). This result was already derived in Ref. [20]within the WKB approximation, where the line has slopeone. It is interesting to note that this result is not ex-actly reproduced in the present calculations, where theCoulomb interaction is taken into account.From the fact that the scaling direction is essentiallya line in the ( ν n , Q )-plane along which the transitionprobability is maximal, we can identify the ratio Q /ν n ≡ γ as a scaling variable. We fitted the maxima along thescaling direction using a q -Gaussian form: T ( ν n , Q ) := T q,β ( Q /ν n ) = 1 √ βC q exp q (cid:0) − βQ /ν n (cid:1) . (9)In the limit of large momenta we find:lim Q →∞ T q,β ( Q /ν n ) (cid:12)(cid:12)(cid:12)(cid:12) ν n = Q /γ ∝ γ / (1 − q ) , (10)i.e. the q -Gaussian takes the form of a power-law, whichis characterized by scale invariance. In Fig. 4(a) we FIG. 4. (a) The maxima of the transition probabilities cor-responding to each transferred momentum Q are plotted fordifferent confining potential strengths η . Dots are numericaldata, while the red lines show the fit with a q -Gaussian distri-bution. (b) By scaling all maxima the transition probabilitiescollapse onto a single curve. show the q -Gaussian fits to the maxima of the transi-tion probability for different confining potentials, i.e.,different values of η . In the limit of large momentumtransfer, Q → ∞ , we obtain, for all η values, powerlaws along the “scaling direction” with a particular scal-ing exponent. Due to the scaling property they are allequivalent up to constant factors. This behavior is pre-sented in Fig. 4(b) where the q -logarithm of the transitionprobability is plotted against the scaling variable Q /ν n .Here ln q is the q -analog of the logarithm defined by:ln q x := ( x − q − / (1 − q ). This plot is seen to produce anessentially linear relation between the scaled transitionprobabilities, which collapse onto a single curve. Devia-tions are due to finite-size effects and numerical precisionin the high Q -regime.The q and β values for the fits in Fig. 4 are shownin fig. 5(a),(b). We see that q and β increase linearly FIG. 5. (a)/(b) The dependence of q / β for different η values.The red line is a guide to the eye. for growing η values. With increasing η the confiningpotential becomes steeper as the interparticle distanceincreases. Therefore one expects that the wave function(eigenfunction of H ) is more localized, which leads to aslower decay of the transition probabilities at high mo-menta. IV. DISCUSSION The results presented in this paper were initiated bythe question whether, and under what conditions, themomentum distribution of a Coulomb systems showsscaling behavior. Following Kimball’s approach [18, 19],we computed the momentum distribution of two inter-acting electrons by numerically solving the two-particleSchr¨odinger equation for a repulsive Coulomb interac-tion in the presence of a confining potential. We foundthat n ( k ) can be parametrized by a q -Gaussian in theentire momentum range. This crossover region connectsthe low-momenta region, described by an ordinary Gaus-sian momentum dependence, with a power-law behaviorat large momenta. In the confinement dominated (in-termediate) momentum region we used the method ofElitzur and Susskind [20] and demonstrated that bound-state resonances also show scaling behavior. In particu-lar, we demonstrated that q -Gaussians are suitable scal-ing functions for the maxima of the transition proba-bility. Indeed, the q -Gaussian behavior is expected toenter in this investigation since it is the natural mathe-matical function that can describe fat-tail distributions,whose asymptotic momentum dependence is not expo-nential but is described, for example, by a power law.Whenever the Coulomb interaction dominates the con-fining potential (in the large momenta region) our resultsrecover the exact analytic results obtained by renormal-ization group techniques [6, 16, 17]. It would be desirableto gain insight into the numerically derived scaling prop-erties also within an analytic approach.Using density functional theory (DFT) [28–31] in com-bination with the impulse approximation we recentlyshowed that Compton profiles of the first column ele-ments of the periodic table can be collapsed onto a singlecurve [22] which can be fitted by a q -Gaussian with el-ement specific ( q, β )-parameters. In that study we didnot address the questions of why there should be scalingbehavior at all, and why the q -Gaussian was found tobe a suitable scaling function. In view of the fact thatin the electronic band theory of solids the periodic ionicpotential provides a natural confining potential, the re-sults of the present paper may provide an explanation ofthe unexpected scaling behavior of the Compton profilesof the alkali elements [22].For the application of the DFT [28–31] in the frame-work of the Kohn-Sham ansatz the knowledge of theexchange-correlation functional is crucial. A centralquantity is the so-called coupling constant integratedpair-correlation function. It accounts for the electroniccorrelations contribution into the kinetic energy [30] andis the input to the derivation of the gradient correctedfunctionals [32, 33]. The contribution of electronic cor- relations to the kinetic energy has been analyzed also inmomentum space [34], and the limits of large and smallmomenta were discussed and found to be in agreementwith results of Kimball [18, 19]. 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