Mean field approximation for the dense charged drop
aa r X i v : . [ c ond - m a t . o t h e r] D ec Mean field approximation for the dense charged drop
S. Bondarenko ∗ , K. Komoshvili † Ariel University, Israel
May 15, 2018
Abstract
In this note, we consider the mean field approximation for the description of the probe chargedparticle in a dense charged drop. We solve the corresponding Schr¨odinger equation for the dropwith spherical symmetry in the first order of mean field approximation and discuss the obtainedresults. ∗ The author declares that there is no conflict of interest regarding the publication of this paper. † The author declares that there is no conflict of interest regarding the publication of this paper.
Introduction
Collisions of relativistic nuclei in the RHIC and LHC experiments at very high energies led to thediscovery of a new state of matter named quark-gluon plasma (QGP). At the initial stages of thescattering, this plasma resembles almost an ideal liquid whose microscopic structure is not yet wellunderstood [1, 2, 3, 4, 5, 6, 7, 8, 9]. The data obtained in the RHIC experiments is in a goodagreement with the predictions of the ideal relativistic fluid dynamics, [10, 11, 12], that establishesfluid dynamics as the main theoretical tool to describe collective flow in the collisions. As an input tothe hydrodynamical evolution of the particles it is assumed that after a very short time, τ < f m/s [12], the matter reaches a thermal equilibrium and expands with a very small shear viscosity [13, 14].In this paper we continue to develop the model proposed in [15]. Namely, we assume that atthe local energy density fluctuations, hot drops [16, 17, 18], are created at the very initial stage ofinteractions at times τ < . f m . These fireballs (hot drops) are very dense and small, their size ismuch smaller than the proton size, see [15], and their energy density is much larger than the densityachieved in high-energy interactions at the energies √ s < GeV . In our model, we assume thatthe fireballs consist of the particles with weak inter-particle interactions and have a non-zero charge.This drop of charged particles we consider from the point of view of quantum statistical physics. Themost general Hamiltonian for this system can be written in the form which describes all possibleinteractions between the particles in the drop: H = H + V + V + . . . + V i + . . . , (1)see [21], where as usual V is an energy of interaction of the particle with an external field, the V isthe energy of pair-like interactions and etc. The mean field approximation for the probe particle in thesystem of charged particles, therefore, can be introduced by the following perturbative scheme. Firstof all, we can consider the motion of only one probe particle in the mean field of all other particles,that corresponds to preserving only V term in the Eq. (1) expression. This approximation will leadto the modification of the propagator of the particle, namely from a free propagator to some ”dressed”one. At the next step, we can take two probe particles, each of them will propagate in the meanfield of the other charges of the system, similarly to the first approximation, but additionally we canintroduce the interaction of these two particles one with another in the mean field of the remainingcharges in the drop, that requires introduction of one V term in Eq. (1) expression in the mean fieldapproximation. Further, we can increase the number of the probe particles in the system, considering,additionally to pair interactions, the interactions of free probe particles and so on.In present calculations, we limit ourselves to the first order of the mean field approach, namely wewill consider the motion of one non-relativistic probe particle in the external mean field created by allother particles in the charged drop. In the absence of the external field, we write the Hamiltonian of the system of charged particles as H = − m Z Ψ + α ( t, ~r ) ∆ Ψ α ( t, ~r ) d x − µ N ++ 12 Z Ψ + β ( t, ~r ) Ψ + α ( t, ~r ′ ) U ( ~r − ~r ′ ) Ψ α ( t, ~r ′ ) Ψ β ( t, ~r ) d x d x ′ , (2)2ere N = Z Ψ + α ( t, ~r ) Ψ α ( t, ~r ) d x (3)is a particles number operator. Considering the mean field approximation for spherically symmetricalsystem, we introduceΨ + α ( t, ~r ′ ) Ψ β ( t, ~r ) ≈ < Ψ + α ( t, ~r ′ ) Ψ β ( t, ~r ) > = δ α β f ( r, r ′ , r ) (4)as some particles density for the droplet with characteristic size r , here r = | ~r | . The f ( r, r ′ , r )function is a distribution function of the system of interests, it can be correctly determined by writingcorresponding Vlasov’s or Boltzmann’s equations coupled to Eq. (1) system. In our case, we will notconsider a particular form of this function, instead we will discuss it’s form basing on some physicalassumptions only. Therefore, we obtain for the Hamiltonian: H = − m Z Ψ + α ( t, ~r ) ∆ Ψ α ( t, ~r ) d x − µ N ++ 12 Z Ψ + α ( t, ~r ) U ( ~r − ~r ′ ) f ( r, r ′ , r ) , Ψ α ( t, ~r ′ ) d x d x ′ , (5)which represents now the energy of the probe particle in the mean field created by the other particlesof the system. Due to the spherical symmetry of the problem, we expand all the operators in theHamiltonian’s expression in terms of spherical harmonic functions. We have for the two-particlesinteraction potential: U ( ~r − ~r ′ ) = ∞ X l = 0 l X m = − l π q l + 1 θ ( r ′ − r ) r l ( r ′ ) l +1 + θ ( r − r ′ ) r ′ l r l +1 ! Y ∗ lm (Ψ , Φ) Y lm ( ψ, φ ) (6)with Ψ , Φ as spherical angles of r ′ vector, ψ, φ as spherical angles of r vector in some sphericalcoordinate system and θ ( r ) as the step function. Correspondingly, we write the particle-field operatoras Ψ α ( t, ~r ) = ∞ X l = 0 l X m = − l ψ α l m ( t, r ) Y lm ( ψ, φ ) . (7)Using the orthogonality property of the harmonic functions Z Y lm ( ψ, φ ) Y ∗ l ′ m ′ ( ψ, φ ) d Ω = δ l l ′ δ m m ′ , (8)with d x = r dr d Ω, we rewrite the Hamiltonian Eq. (5) in one-dimensional form as a function of r and r ′ only: H = − m ∞ X l = 0 l X m = − l Z ∞ (cid:18) ψ + α l m ( t, r ) ∆ r ψ α l m ( t, r ) − l ( l + 1) r ψ + α l m ( t, r ) ψ α l m ( t, r ) (cid:19) r dr −− µ ∞ X l = 0 l X m = − l Z ∞ ψ + α l m ( t, r ) ψ α l m ( t, r ) r dr ++ ∞ X l = 0 l X m = − l π q l + 1 Z ∞ r dr Z ∞ r r l r ′ ( l + 1 ) f ( r, r ′ , r ) ψ + α l m ( t, r ) ψ α l m ( t, r ′ ) r ′ dr ′ ++ Z r r ′ l r ( l + 1) f ( r, r ′ , r ) ψ + α l m ( t, r ) ψ α l m ( t, r ′ ) r ′ d r ′ ! . (9)In the next Section, we solve Schr¨odinger equation corresponding to this Hamiltonian.3 Equations of motion
We introduce the usual commutation relations for the fields of interest (see Eq. (7)): { Ψ α ( t, ~r ) , Ψ + β ( t, ~r ′ ) } = δ α β δ ( ~t − ~r ′ ) . (10)Using Eq. (8) property, we correspondingly obtain one-dimensional commutation relations for the newfields: { ψ α l m ( t, r ) , ψ + β l ′ m ′ ( t, r ′ ) } = 1 r δ α β δ l l ′ δ m m ′ δ ( r − r ′ ) . (11)The Schr¨odinger equation for ψ α l m ( t, r ) field, therefore, has the following form: ı ∂∂t ψ α l m ( t, r ) = (cid:18) − m (cid:18) ∆ r − l ( l + 1) r (cid:19) − µ (cid:19) ψ α l m ( t, r ) ++ 4 π q l + 1 Z ∞ r r l r ′ ( l + 1 ) f ( r, r ′ , r ) ψ α l m ( t, r ′ ) r ′ dr ′ ++ Z r r ′ l r ( l + 1) f ( r, r ′ , r ) ψ α l m ( t, r ′ ) r ′ d r ′ ! . (12)Rescaling the drop’s density function and rewriting it in the dimensionless form as f ( r, r ′ , r ) → r / ( r ′ ) / f ( r, r ′ , r ) , (13)introducing new variable in integrals in Eq. (12) r ′ = x r , (14)we rewrite the integrals in Eq. (12) finally as4 πr ( 2 l + 1 ) (cid:18)Z ∞ dxx l + 1 / f ( r, x, r ) ψ α l m ( t, x ) + Z dx x l + 1 / f ( r, x, r ) ψ α l m ( t, x ) (cid:19) . (15)For the case of the drop of small size, we can expand our ψ function in Eq. (15) around x = 1 in bothterms, this point gives the main contribution to both integrals. Therefore, in the first approximation,we have for Eq. (15):4 πr ( 2 l + 1 ) (cid:18)Z ∞ dxx l + 1 / f ( r, x, r ) ψ α l m ( t, xr ) + Z dx x l + 1 / f ( r, x, r ) ψ α l m ( t, xr ) (cid:19) ≈ ψ α l m ( t, r ) 4 πr ( 2 l + 1 ) (cid:18)Z ∞ dxx l + 1 / f ( r, x, r ) + Z dx x l + 1 / f ( r, x, r ) (cid:19) == Q l ( r ) r ψ α l m ( t, r ) , (16)with Q l as a l multipole moment of the drop. The Schr¨odinger equation Eq. (12) now acquires thefollowing form: ı ∂∂t ψ α l m ( t, r ) = − m (cid:18) ∆ r − l ( l + 1) r (cid:19) − µ + q r Q l ( r ) ! ψ α l m ( t, r ) . (17)Representing the wave function as ψ α l m ( t, r ) = u α l m ( r ) e − ı t ( E − µ ) (18)4e obtain the Schr¨odinger equation for the particle in the following form: m ∆ r − m l ( l + 1) r + E − q r Q l ( r ) ! u α l m ( r ) = 0 . (19)In general, we can not solve this equation without knowledge of the form of f ( r, x, r ) particlesdistribution function in Eq. (15) integrals . Nevertheless, we can guess the form of the function in the r ≤ r region of the drop, mostly interesting for us. Indeed, at r ≫ r , which is outside of the dropregion, the potential Eq. (16) is the usual Coulomb potential, but in the r ≤ r region, the situation isdifferent. The existence of the drop requires the presence of some potential well at r ≤ r which willkeep particles inside the drop during some (very short) time and, therefore, it must be the potential’sminimum present somewhere between r = 0 and r ∝ r . Hence, this minimum is the indication ofthe creation of the dense drop of finite size in the interaction system of interest and, consequently, wecan write the potential energy from Eq. (19) in this region as12 m l ( l + 1) r + Q l ( r ) q r ≈ m l ( l + 1) r min + Q l ( r min ) q r min + A l ( r min ) q r ( r − r min ) , (20)where we assumed that the potential energy acquires it’s minimum at r min , the A l ( r min ) here arethe positive coefficients of the potential’s expansion around this minimum. This situation, in fact, issimilar to the situation in the system of two-atoms molecules, see [22] and thereafter, where two atomsare kept inside some mutual potential well. Inserting Eq. (20) expansion in Eq. (19), we obtain thefollowing equation: m ∆ r + E − m l ( l + 1) r min − Q l ( r min ) q r min − A l ( r min ) q r ( r − r min ) ! u α l m ( r ) = 0 . (21)The solution of this equation is similar to the solution of the Schr¨odinger equation for the harmonicoscillator with energy levels defined by E ′ = E − m l ( l + 1) r min − Q l ( r min ) q r min , (22)and consequently the energy levels of the system are given by E n l = 12 m l ( l + 1) r min + Q l ( r min ) q r min + q (cid:18) A l ( r min ) m r (cid:19) / ( n + 1 / n = 0 , , , . . . , (23)with the wave functions u α l m ( r ) = c α l m e − ( r − r min ) / (2 R ) r √ R H n (( r − r min ) / R ) , (24)where R = r m A l ( r min ) q ! / . (25)We note, that the obtained solution is indeed similar to the solution of Schr¨odinger equation for thetwo-atoms molecules, see [22] for example. 5 Conclusion
In this note, we demonstrate that the spectrum of the charged non-relativistic particle in the densecharged drop has a quantum structure, see Eq. (23) and it is determined by three terms in the firstmean field approximation. The first term in Eq. (23) can be considered as the quantized rotation energyof the drop, the second one is the quantized electrostatic energy due to the miltipole moments of thecharged drop. The third term in Eq. (23) expression is the usual quantum corrections to the energydue to the oscillation of the particle inside the drop with eigenfrequencies determined by the form ofthe distribution function of the particles in the drop. The presence of this minimum is a necessarycondition of the drop’s creation, see also [15]. The values of these corrections have an additionaldegeneracy of energy levels defined by l quantum number in comparison to the ordinary quantumoscillator. In this formulation, the considered problem is similar to the problem of the description ofsystem of two-atom molecule, see [22] . The further development of the approach can include theconsideration of higher orders of mean mean field approximation for the system and introducing thekinetic equation for the Eq. (4) distribution function coupled to the Hamiltonian, these problems weplan to consider in the following publications.We conclude, that our model can be useful for the clarification of the spectrum of the producedparticles which is influenced by the quantum-mechanical properties of the QCD fireball. We believe,that this approach will provide the connection between the data, obtained in high-energy collisionsof protons and nuclei in the LHC and RHIC experiments [10, 11, 12, 19, 20], and microscopic fieldsinside the collision region. We note, that the proposed approach can be used also for the description of bound states created at low energyinteractions as well, we plan to investigate this subject in a separate publication. eferences [1] E.Shuryak, Nucl. Phys. B
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