Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil
Haas, Fernando An Introduction to Quantum Plasmas Brazilian Journal of Physics, vol. 41, núm. 4-6, 2011, pp. 249-263 Sociedade Brasileira de Física Sâo Paulo, Brasil
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How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Braz J Phys (2011) 41:349–363 DOI 10.1007/s13538-011-0043-0 GENERAL AND APPLIED PHYSICS
An Introduction to Quantum Plasmas
Fernando Haas
Received: 26 August 2011 / Published online: 16 September 2011 © Sociedade Brasileira de Física 2011
Abstract Shielding effects in non-degenerate and de- Quantum mechanics becomes relevant in plasmas when generate plasmas are compared. A detailed derivation the de Broglie wavelength of the charge carriers is of the Wigner-Poisson system is provided for electro- comparable to the inter-particle distance, so that there static quantum plasmas in which relativistic, spin, and is a significant overlap of the corresponding wave- collisional effects are not essential. A detailed deriva- functions. In such a situation, the many-body problem tion of a quantum hydrodynamic model starting from is described by Fermi-Dirac statistics, in contrast to the Wigner-Poisson system is presented. The route for the usual laboratory and space plasmas, which obey this derivation considers the eikonal decomposition of Maxwell-Boltzmann statistics. The fermionic charac- the one-body wavefunctions of the quantum statistical ter becomes prominent for sufficiently dense plasmas. mixture. The merits and limitations of the resulting Among these, one can cite plasmas in such compact as- quantum hydrodynamic model are discussed. trophysical objects as white dwarfs and the atmosphere of neutron stars [5], or in the next generation intense- Keywords Quantum plasmas · Fermion systems · laser–solid-density plasma interaction experiments [6]. Electron gas · Quantum transport · Plasma kinetic Moreover, many-body charged particle systems can- equations not be treated by pure classical physics when the char- acteristic dimensions become comparable to the de Broglie wavelength. This is the case for quantum semi- 1 Introduction conductor devices, like high-electron-mobility transis- tors, resonant tunneling diodes, or superlattices. The Plasma physics is frequently not a part of the standard operation of these ultra-small devices relies on quan- physics curriculum. Thus, theoretical physicists who tum tunneling of charge carriers through potential bar- study quantum mechanics are not usually exposed to riers. For these systems both Maxwell-Boltzmann or plasma science and have no interest in it. And vice Fermi-Dirac statistics can be applied, according to the versa, plasma physicists often do not have the required particle density. For instance, the Fermi-Dirac charac- background in quantum theory. ter should be taken into account in the drain region + + Plasma physics is commonly thought of as a purely of n nn diodes [7, 8]. In addition, due to recent classical field. Nonetheless, the last ten years have advances in femtosecond pump-probe spectroscopy, witnessed renewed interest in plasma systems marked quantum plasma effects are attracting attention in the by quantum effects; see [1–4] for extensive reviews. physics of metallic nanostructures and thin metal films [9]. Also, X-ray Thomson scattering in high energy density plasmas provide experimental techniques for B accessing narrow bandwidth spectral lines [10], so as F. Haas ( ) to detect frequency shifts due to quantum effects. Such Departamento de Física, Universidade Federal do Paraná, 81531-990, Curitiba, Paraná, Brazil modifications of the plasmon dispersion relation are e-mail: [email protected] inline with theoretical predictions [11]. In this context, 350 Braz J Phys (2011) 41:349–363 precise experiments have been suggested [12]inorder charge. Eventually, in the equilibrium situation, a sta- to measure low-frequency collective oscillations (ion- tionary cloud of negative charge would accumulate acoustic waves) in dense plasmas, as soon as keV free around qt. Then, instead of qt an external observer electron lasers become available. would see an effective, smaller shielded charge. This is The purpose of this short introduction is two-fold. the (static) screening, or shielding effect in plasmas, in First, we present an elementary discussion on the itself a manifestation of the quasi-neutrality property: screening of a test charge in non-degenerate and dege- due to the electric force, any excess charge tends to be nerate plasmas. Shielding is one of the most emblematic compensated. In the stationary regime, the electrostatic collective effects in plasmas, so that it is appropriate field φ = φ(r) is described by Poisson’s equation, to comment on the differences between its classical ∇2φ = e ( ) − − qt δ( ), and quantum regimes. In this way, a convenient intro- n r n0 r (1) ε0 ε0 duction to quantum plasmas is provided. Second, we where −e < 0 is the electron charge, ε is vacuum per- consider the basic kinetic and fluid models for quantum 0 mittivity, and for definiteness the test charge is located plasmas. We hence present the Wigner-Poisson system, at the origin. For simplicity we assume the test charge which is the quantum equivalent of the Vlasov-Poisson to be sufficiently massive that it can be considered system in classical plasma physics. Next, we discuss the at rest. derivation of fluid equations from the Wigner-Poisson Assuming a Maxwell-Boltzmann statistics, the qui- model. Special attention is paid to the transition from escent electron gas being in thermodynamical equilib- microscopic to macroscopic descriptions. While none rium at a temperature T, one would have of these subjects is new, most of the calculations here are intended to be shown in more detail than the eφ n(r) = n0 exp , (2) usual discussion in the literature. In an attempt to κBT restrict the treatment to a basic level, only electrostatic, where κ is Boltzmann’s constant. weakly coupled and non-relativistic quantum plasmas B The Maxwell-Boltzmann statistics is appropriate for are considered. Hence, spin, magnetic field, collisional dilute, or non-degenerate plasmas, for which the degen- or relativistic effects are not addressed. eracy parameter χ = T /T 1. Here T = E /κ is This work is organized as follows. In order to intro- F F F B the Fermi temperature, defined in terms of the Fermi duce the basic physical parameters for non-degenerate energy [13] and degenerate plasmas, Section 2 discusses screening 2 in these systems. Section 3 introduces the basic ki- / E = (3π 2n )2 3, (3) netic model for electrostatic quantum plasmas, namely F 2m 0 the Wigner-Poisson system. Some of the fundamen- where is Planck’s constant over 2π and m is the tal properties of the Wigner function are analyzed, electron mass. Since electrons are (spin-1/2) fermions, the Wigner function playing the rôle of a (quasi)- even in the limit of zero thermodynamic tempera- probability distribution in phase space. Section 4 ap- ture it is impossible to accommodate all of them in plies an eikonal, or Madelung decomposition of the the ground state, due to the Pauli exclusion principle. one-body wavefunctions defining the quantum statis- Hence, excited single-particle states are filled up until tical ensemble. In this way the pressure functional is the highest energy level, whose corresponding energy is shown to be the sum of two terms, associated with the defined as EF . kinetic and osmotic velocity dispersions (to be defined Assuming the scalar potential to be zero before the later), plus a Bohm potential term associated to quan- insertion of the test charge, we can linearize (1)to tum diffraction effects. Section 5 is dedicated to final obtain remarks. n e2 q ∇2φ = 0 φ − t δ(r). (4) ε0κBT ε0 2 Shielding in Non-degenerate and Degenerate The radially symmetric solution to (4) with appropriate Plasmas boundary conditions is the Yukawa potential
qt − /λ φ = e r D , (5) Suppose a test charge qt > 0 is added to a plasma 4πε0r composed by an electron gas [of number density n(r)] where and a fixed homogeneous ionic background (of num- ε κ 1/2 ber density n0). Initially, due to the Coulomb force, λ = 0 BT D 2 (6) the electron trajectories would deviate toward the test n0e Braz J Phys (2011) 41:349–363 351
is the (electron) Debye length. From (5), an observer ing distance λF , or Thomas-Fermi length [3], can be set would measure a very small potential for distances for degenerate plasmas, larger than the Debye length, which is inline with the 1/2 2ε0 EF linearization procedure. Therefore λD is a fundamental λ = . F 2 (12) length for dilute, non-degenerate plasmas, playing the 3n0e rôle of an effective range of the Coulomb interaction. Unlike the Debye length λD, the Thomas-Fermi length Screening is a collective effect due to a large number of is nonzero even at zero thermodynamic temperature. electrons around qt. This is due to the exclusion principle, which prevents What happens in the degenerate case? To start an- the accumulation of electrons at the position occupied swering, notice that the number density (2) follows by the test charge. from the zeroth-order moment of the local Maxwell- Notice that in a sense every particle in a plasma, be Boltzmann one-particle equilibrium distribution func- it degenerate or not, can be interpreted as a test charge = ( , ) tion fcl fcl r v given by with the corresponding screening cloud. Hence instead of the long-range Coulomb field we have an effective / m 3 2 1 mv2 Yukawa interaction field. This point of view is adopted, fcl(r, v) = n0 exp − − eφ . 2πκBT κBT 2 for instance, in the treatment of the ultrafast phase- space dynamics of ultracold, neutral plasmas [15]. (7) In our simplified picture fcl(r, v) was regarded as In other words, a purely classical probability distribution function. A more detailed treatment taking into account quantum n(r) = dv fcl(r, v). (8) diffraction effects shows that at large distance the inter- particle oscillation behaves as On the other hand, the simplest approach for a 1 φ ∼ cos(2r/λ ), (13) degenerate plasma would consider an uniform distri- r3 F bution of electrons for energy smaller than the Fermi energy, and no particles above the Fermi level. Then a phenomenon known as Friedel oscillations [16, 17]. (7) could be replaced by As discussed in the next Section, the Wigner function ⎧ provides a convenient tool to incorporate quantum 2 ⎨⎪ 3n0 mv effects in plasmas, in strict analogy with the classical if − eφ to restore complete charge neutrality. The resulting fcl = fcl(x,v,t) is the reduced one-particle probability linear oscillations have frequency ωp. distribution function. For a plasma with N electrons, • Typical interaction energy Uint for both classical (1/N) fcl(x,v,t) dv dx gives the probability of finding and quantum plasmas: one electron with position between x and x + dx and velocity between v and v + dv,atthetimet.The 2 1/3 = e n0 , normalization Uint ε (16) 0 since the mean inter-particle distance scales as dv dxfcl(x,v,t) = N (21) −1/3 n0 . • Typical kinetic energies: K = κ T for non- C B is hence assumed. degenerate and K = κ T for degenerate plas- Q B F Once the Vlasov-Poisson system subject to appro- mas. The large fermionic kinetic energies are due to priate boundary conditions is solved, either analytically the Pauli exclusion principle, which forces excited or numerically, to determine f , the whole machinery states to be filled even at zero temperature. cl of classical statistical mechanics becomes available to From the typical energy scales we can form classical compute the expectation values of macroscopic quan- C and quantum Q energy coupling parameters, tities. For example, the average kinetic energy of one / / electron follows from U e2n1 3 n1 3 = int = 0 = 2.1 × 10−4 0 , (17) C ε κ 2 2 KC 0 BT T mv 1 mv < >= dv dxf (x,v,t) . (22) 2 cl U 2me − / 2 N 2 = int = = 5.0 × 1010 n 1 3 . (18) Q / 1/3 0 KQ (3π 2)2 3ε 2n 0 0 It is convenient to adopt a similar methodology in a The numerical values were calculated for S.I. units. quantum kinetic theory for plasmas, as far as possible. Weakly coupled plasmas have small energy coupling The Wigner function f = f(x,v,t), a quantum parameters. From (17–18) the conclusion is that while equivalent of fcl(x,v,t), affords a phase-space formula- classical weakly-coupled plasmas tend to be dilute tion of quantum mechanics. Let us study some of its ba- and cold, quantum weakly-coupled plasmas tend to be sic properties. For an one-particle pure state quantum 35 −3 dense. For example, with n0 > 10 m (white dwarf) system with wavefunction ψ(x, t), the Wigner function one has Q < 0.1, allowing the use of collisionless mod- is defined as [19] els in a first approximation. This is a consequence of the Pauli exclusion principle, which tends to bar e-e v collisions in very dense systems. In other words, the = m im s ψ∗ + s , ψ − s , , f π ds exp x t x t more dense a degenerate plasma is, the more it resem- 2 2 2 bles an ideal gas, except for the mean field, collective (23) interaction. A more detailed account on the characteristic scales with all symbols as before. From f(x,v,t) we can com- in classical and quantum Coulomb systems can be pute the probability density found in [1, 4, 18]. dv f(x,v,t) =|ψ(x, t)|2 (24) 3 Obtaining the Wigner-Poisson System and the probability current The Vlasov-Poisson system ∂ ∂ ∂φ ∂ i ∂ψ∗ ∂ψ fcl fcl e fcl v ( ,v, )v = ψ − ψ∗ . + v + = 0, (19) d f x t ∂ ∂ (25) ∂t ∂x m ∂x ∂v 2 m x x ∂2φ e = dv f (x,v,t) − n (20) The wavefunction ψ(x) is normalized to unity. ∂ 2 ε cl 0 x 0 In general, even a one-particle quantum system can- is the basic tool in the kinetic theory of classical not be represented by a single wavefunction. Mixed plasmas, where for simplicity we have considered a quantum states are described by a quantum statistical one-dimensional electron plasma in a fixed neutral- ensemble {ψα(x, t),pα},α = 1, 2, ...M, with each wave- izing ionic background n0. In the Vlasov Eq. (19), function ψα(x, t) having an occupation probability pα Braz J Phys (2011) 41:349–363 353 ≥ M = such that pα 0,with α=1 pα 1. The Wigner func- particles have the same mass m.TheN-particle Wigner tion is then defined by the superposition function is defined by f N(x ,v...,, x ,v , t) M v 1 1 N N m im s f = pα ds exp M N π m N im = visi 2 α= = ... i 1 1 N π pα ds1 dsN exp 2 α= ∗ s s 1 ×ψα x + , t ψα x − , t . (26) 2 2 N ∗ s1 sN × ψα x + , ..., x + , t 1 2 N 2 N s1 sN Correspondingly, we have the probability and current × ψα x1 − , ..., xN − , t . (31) densities 2 2 The factor N in (31)isinsertedsothat M 2 dv f(x,v,t) = pα |ψα(x, t)| , (27) N α=1 dx1dv1...dxNdvN f (x1,v1, ..., xN,vN, t) = N. (32) M ∂ψ∗ ∂ψ v ( ,v, )v = i ψ α −ψ∗ α . d f x t pα α ∂ α ∂ (28) In this manner, the integral of f N over all the velocities 2 m α= x x 1 gives a number density in configuration space. The fermionic character of the system is not necessarily The density matrix ρ(x, y, t) could also be elected as included, but this can be done by assuming the N-body the central object in a quantum kinetic theory for plas- ensemble wavefunctions to be antisymmetric in (31). mas, but in this approach the similarity to the Vlasov- Wigner functions provide a convenient mathemat- Poisson model would be lost. Besides, the formulation ical tool to calculate average quantities, and hence defined by (26–28) is in complete correspondence to the are analogous to the classical probability distribution density-matrix formalism, since function. Nonetheless, they are not necessarily positive definite. Hence it is customary to refer to them as quasi- probability distributions. M ∗ To proceed to the derivation of the quantum analog ρ(x, y, t) ≡ pα ψα(x, t)ψα (y, t) α=1 of the Vlasov-Poisson system (19)and(20), we can v( − ) + introduce the reduced one-particle Wigner function im x y x y ( ,v , ) = dv exp f ,v,t , (29) f x1 1 t , 2 ( ,v , ) = v ... v N( ,v , ..., ,v , ), with inverse given by f x1 1 t dx2d 2 dxNd N f x1 1 xN N t (33) m imv s s s f(x,v,t)= ds exp ρ x + , x − , t . (2)( , 2 π 2 2 and the reduced two-particle Wigner function f x1 v1, x2,v2, t), with a convenient normalization factor N, (30) (2) N f (x1,v1, x2,v2, t) = N dx3dv3...dxNdvN f In other words, given the Wigner function the density matrix can be found and vice-versa. The same applies × (x1,v1, ..., xN,vN, t). (34) to many-body Wigner functions and density matrices. To proceed one step further, consider now a Similar reduced N-particle Wigner functions with N-particle statistical mixture described by the set N ≥ 3 can be likewise be defined. If the Wigner N {ψα (x1, x2, ..., xN, t),pα}, where the normalized N- function were a true probability distribution, (1/N) N particle ensemble wavefunctions ψα (x1, x2, ..., xN, t) f(x1,v1, t)dx1dv1 would give the probability of finding ,α= , , ..., v ( ,v ) are distributed with probabilities pα 1 2 M sa- the particle 1 in an area dx1d 1 centered at x1 1 , ≥ , M = tisfying pα 0 α=1 pα 1 as before. Here xi repre- irrespective of the “position” and “velocity” of the sents the position of the ith-particle, i = 1, 2, ...N.All remaining ith-particles, i = 2, ..., N. An analogous 354 Braz J Phys (2011) 41:349–363 “classical” probability interpretation could be assigned in both the classical infinite BBGKY set of equations to the remaining reduced Wigner functions, except for and its quantum analogue we are faced with a closure the non-positive-definiteness property. problem. In passing, we notice that Ignoring correlations is the simplest way to close the system, i.e., assuming that the distribution of particles at v ( ,v , )= , dx1d 1 f x1 1 t N (xi,vi) is unaffected by the particles at a distinct phase space point (x j,vj). In this mean field (or Hartree) (2) 2 dx1dv1dx2dv2 f (x1,v1, x2,v2, t) = N . (35) approximation the N-body Wigner function factorizes, (2) It is reasonable to pay special attention to the re- f (x1,v1, x2,v2, t) = f(x1,v1, t) f(x2,v2, t). (39) duced Wigner functions, since f N contain far more information than what is commonly needed. In this Equation (38) then becomes regard the one-particle Wigner function plays a distin- guished rôle. Indeed, macroscopic objects like number ∂ f ∂ f and current densities can be derived from f after inte- + v1 = dv K[Wsc | v − v1, x1, t] f(x1,v , t), ∂ t ∂ x 1 1 1 gration over just one velocity variable, exactly as in (27) 1 and (28), originally written for a one-particle system. (40) To obtain the evolution equation satisfied by the one-body Wigner function, the philosophy of [20]can with the mean field self-consistent potential be pursued. Consider then the Schrödinger equation satisfied by the N-body ensemble wavefunctions, Wsc(x, t) = dv dx f(x ,v,t) W(|x − x |). (41) ∂ψN 2 N ∂2ψ N α =− α + ( , ..., )ψN i V x1 xN α (36) The functional K[W | v − v , x , t] is defined by ∂t 2m ∂ x2 sc 1 1 1 i=1 i [ | v − v , , ] for an interaction energy V(x , ..., x ). K Wsc 1 x1 t 1 N 1 We are concerned with cases in which the system im im(v − v )s =− ds exp − 1 1 1 components interact through some two-body poten- 2π2 1 tial W, s1 s1 × Wsc x1 + , t − Wsc x1 − , t . (42) V(x1, ..., xN) = W(|xi − x j|). (37) 2 2 i< j Frequently an external, possibly time-dependent po- As it is evident, (37) is particularly interesting because tential Vext(x1, ..., xN, t) should be included. Such a cir- it describes the Coulomb interaction. cumstance arises, for instance, in solid-state devices, Some algebra then shows that [1, 20] when the electronic motion in a fixed ionic lattice or ∂ f ∂ f im under a confining field like in quantum wires or quan- + v =− ds dv dx dv 1 2 1 1 2 2 tum wells is considered [7, 8, 25]. Or even the field due ∂ t ∂ x1 2π to a homogeneous ionic background can be regarded im v − v1 s1 × exp − 1 as an external superimposed field. Hence, consider an external potential of the form s1 s1 × W x −x + −W x −x − N 1 2 2 1 2 2 Vext(x1, ..., xN, t) = Wext(xi, t) (43) ( ) × 2 ( ,v , ,v , ), = f x1 1 x2 2 t (38) i 1 which relates the reduced one-particle Wigner func- for some one-particle potential Wext(xi, t). It is implicit tion f to the reduced two-particle Wigner function in (43) that the functional form of W is the same (2) ext f . In the derivation, it was taken into account that irrespective of x , so that the external field has the same i N 1. As a matter of fact, a more detailed argu- influence on all particles. For completeness we record ment involving the higher-order Wigner functions yield here the potential a quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy [21–24], which shows the dynam- N ( − ) ics of the N 1 -body reduced Wigner function to de- V(x1, ..., xN) = W(|xi − x j|) + Wext(xi, t). (44) pend on the N-body reduced Wigner function. Hence i< j i=1 Braz J Phys (2011) 41:349–363 355 Following the same steps as before, the one-body re- case of doped semiconductors, or in the presence of a duced Wigner function f(x1,v1, t) can then be shown dispersive medium with permittivity constant ε = ε0. to satisfy the equation From (50)and(51) it is straightforward to derive the ∂ ∂ equality f + v f ∂ t 1 ∂ x e 1 ∇2φ = ( , , ) − , ε dv f r v t n0 (52) = v [ + | v −v , , ] ( ,v , ), 0 d 1 K Wsc Wext 1 1 x1 t f x1 1 t (45) the Poisson equation pertinent to the case under study. for Notational simplicity recommends that we once more restrict our discussion to the one-dimensional [ + | v − v , , ] K Wsc Wext 1 1 x1 t φ case. In terms of the electrostatic potential ,(45)is im im(v − v )s rephrased as =− ds exp − 1 1 1 2π2 1 ∂ f ∂ f + v = dv Kφ[φ | v − v, x, t] f(x,v, t), (53) s1 s1 ∂ t ∂ x × Wsc x1 + , t + Wext x1 + , t 2 2 where Kφ[φ | v − v, x, t] is the functional s1 s1 −Wsc x1 − , t − Wext x1 − , t (46) 2 2 iem ds im(v − v)s Kφ[φ | v − v, x, t]= exp where the averaged self-consistent potential W is 2π sc again given by (41). s s × φ x+ , t −φ x− , t . (54) The necessary changes in three-dimensional 2 2 charged particle motion are as follows. Consider, Equation (53) can be termed the quantum Vlasov equa- for definiteness, the Coulomb interaction tion (in the electrostatic case), since it is the quantum 2 analog of the Vlasov equation satisfied by the reduced (| − |) = e . W r r (47) one-particle probability distribution function. Finally, 4πε0 |r − r | the quantum Vlasov equation should be coupled to φ( , ) and define the total electrostatic potential r t so that Poisson’s equation, φ( , ) = φ ( , ) + φ ( , ), r t sc r t ext r t (48) ∂2φ e = dv f(x,v,t) − n . (55) ∂ 2 ε 0 the two terms on the right-hand side being associated x 0 with the self-consistent potential Wsc and some external Equations (53)and(55) constitute the Wigner- potential Wext. Explicitly, Poisson system, which is the fundamental kinetic model for electrostatic quantum plasmas. It self-consistently ( , ) =− φ ( , ), ( , ) =− φ ( , ). Wsc r t e sc r t and Wext r t e ext r t determines both the Wigner function, associated with (49) the particle distribution in phase space, and the scalar potential, which in turn describe the forces on the From the three-dimensional version of (41)itfol- particles. lows that The Wigner-Poisson system needs to be supple- 1 mented with suitable boundary and initial conditions. ∇2φ =−e ( , , ) ∇2 sc dv dr f r v t ε0 4π |r − r | For plasmas, decaying or periodic boundary conditions are frequently employed. For nanodevices, the choice = e ( , , )δ( − ) ε dv dr f r v t r r of boundary conditions is subtler due to the finite size 0 of the system and the nonlocal character of the Wigner e = dv f (r, v, t). (50) function. Indeed, to compute the integral defining the ε0 Wigner function we need to specify f(x,v,0) over the Moreover, entire space, even when dealing with finite size systems [7, 8, 25]. 1 n e ∇2φ =− ∇2 ≡− 0 It seems appropriate, at this point, to recapitulate ext Wext ε (51) e 0 the simplifications in the derivation of the Wigner- if the external potential is due to an immobile fixed Poisson system. Above all, it is a mean field model, homogeneous ionic background of density n0 and ion with the N-body ensemble Wigner function assumed to charge e. Appropriate changes are needed to describe be factorisable, in order to achieve the simplest closure a non-homogeneous background [7, 8, 25], e.g., in the of the quantum BBGKY hierarchy. Hence the N-body 356 Braz J Phys (2011) 41:349–363 Schrödinger equation (or the Liouville-von Neumann along the (classical) characteristic equations equation for the N-body ensemble density matrix) is dx dv e ∂φ replaced by a system with fewer degrees of freedom. = v, = . ∂ (58) Indeed, for N electrons in three-dimensional space, we dt dt m x have 3N + 1 coordinates to define the wavefunction, It follows that the positive definiteness of the Wigner 6N + 1 coordinates for the density matrix, and only 6 + function is not preserved by (53), except for linear 1 = 7 independent variables for the reduced one-body electric field and vanishing quantum correction. Also Wigner function, time taken into account here. The notice that, although similar to, (57) is not truly a mean field theory is, therefore, much less numerically Boltzmann equation, since it remains invariant un- demanding, since it requires discretization of a space der time-reversal t →−t, x → x,v →−v, f(x,v,t) → with fewer dimensions. The price for the reduction is f(x, −v,−t) (assuming that φ(x, t) = φ(x, −t)). There the neglect of collisions, besides—at least in the present is no irreversibility nor memory loss at all in the formulation—spin and relativistic effects. And finally, quantum Vlasov equation, be it in the semiclassical or no magnetic fields were included. the fully quantum versions. This is hardly surprising, Since it is the analog of the Vlasov-Poisson, the since the Schrödinger equation is invariant under time Wigner-Poisson system becomes the natural tool in reversal. quantum kinetic theory for electrostatic plasmas. With Even when the quantum Vlasov and Vlasov equa- some optimism, the methods traditionally applied to tions are the same, as in the case of linear electric fields, the Vlasov-Poisson system can be translated to Wigner- f(x,v,t) cannot be viewed as an ordinary probability Poisson quantum plasmas. Nevertheless, other quan- distribution function. Not all phase-space functions are tum kinetic treatments of charged particle systems acceptable as Wigner functions, since a genuine Wigner are obviously important. For example, the density- function corresponds necessarity to a positive definite functional [25] and Green’s function [26, 27]ap- density matrix. At least the following necessary condi- proaches are popular tools for the modeling quantum tions hence emerge [28], transport in the condensed matter community. More- over, alternative approaches can sometimes overcome dxdv f = N, (59) the simplifications of the Wigner-Poisson model. For instance, Green’s function techniques can be used to dv f ≥ 0, (60) describe collisions associated to short range particle- particle interactions [26, 27], in terms of a Boltzmann dxf ≥ 0, (61) type collision operator. It is instructive to examine the semiclassical limit of mN2 the quantum Vlasov (53). Combined with the change dxdv f 2 ≤ . (62) 2π of variable s = τ/m, a Taylor expansion leads to the approximate equality Equation (59) is simply a normalization condition, while (60)and(61) assure the spatial and velocity marginal probability densities to be everywhere non- ∂ f ∂ f e ∂φ ∂ f e2 ∂3φ ∂3 f + v + = + O(H4). negative. Equation (62) eliminate excessively spiky ∂ ∂ ∂ ∂v 3 ∂ 3 ∂v3 t x m x 24 m x Wigner functions, which would violate the uncertainty (56) principle. For instance, for the Gaussian profile N x2 v2 Implicitly, the semiclassical approximation assumes the f = exp − exp − (63) πσ σ σ 2 σ 2 smallness of a non-dimensional quantum parameter 2 x v 2 x 2 v H = /(mv L ),wherev and L are characteristic 0 0 0 0 with constant standard deviations σx,v, it follows from velocity and length scales, respectively. (62)that Equation (56) is a semiclassical Vlasov equation, with f playing the rôle of one-particle distribution σxσv > , (64) function. We see that unlike the result for classical 2m plasmas, in general neither f nor phase space volume in accordance with the uncertainty principle. are preserved by the quantum Vlasov equation, since Since no collisional effects are included in the Wigner-Poisson system, strongly coupled quantum df e2 ∂3φ ∂3 f plasmas require special treatment. Hence, in princi- = + O(H4) = 0, (57) dt 24m3 ∂ x3 ∂v3 ple, the Wigner-Poisson model assumes a small energy Braz J Phys (2011) 41:349–363 357 coupling parameter; see (17)and(18). For dense plas- it can be shown that the pressure term appearing in mas (such as the electron gas in metals), however, quan- the fluid equations can be separated into a classical tum collisionless models are sometimes still applicable and a quantum parts. A working hypothesis is then thanks to the Pauli blocking of e-e collisions [4], even if applied to the classical term, so as to close the fluid the energy coupling parameter is large. system. Shortly, the meaning of classical and quantum Not only very dense charged particle systems re- contributions to the pressure will be explained. quire quantum kinetic equations. For instance, due to This approach to a quantum hydrodynamic model the ongoing miniaturization, even scarcely populated for plasmas appeared in [30]. The same quantum fluid electronic systems such as resonant tunneling diodes model has been applied to several distinct problems should be described by quantum models [7]. Indeed, involving charged particle systems, for instance, the the behavior of these ultra-small electronic devices re- nonlinear electron dynamics in thin metal films [9], the lies on such quantum diffraction effects as tunneling, excitation of electrostatic wake fields in nanowires [31], which makes purely classical methods inappropriate. parametric amplification characteristics in piezoelectric The non-local integro-differential potential term in (53) semiconductors [32], breather waves in semiconductor in the Wigner-Poisson system has been shown to be quantum wells [33], multidimensional dissipation-based able to model the negative differential resistance as- Schrödinger models from quantum Fokker-Planck dy- sociated to tunneling [29]. Moreover, in view of the namics [34], the description of quantum diodes in de- nanometric scale of the devices, the collisionless ap- generate plasmas [35] and quantum ion-acoustic waves proximation becomes more reasonable, simply because in single-walled carbon nanotubes [36], to name but a the mean free-path exceeds the system size. few. The model was extended to incorporate magnetic In the same manner, the usually extreme high op- fields in [37]. erating frequencies makes the collisionless approxima- We take moments of (53) by integrating over the tion more accurate, because ωτ 1 for typical fre- velocity space. In other words, introducing the standard quency ω and average time τ between collisions. For definitions of density, mean velocity and pressure example, in resonant tunneling diodes potential barri- 1 ers of the order 0.3 eV ∼ ω are often found [7], which n(x, t) = fdv, u(x, t) = fv dv, n correspond to an operating frequency ω ∼ 1015 s−1. The Wigner-Poisson system is therefore well suited P(x, t) = m fv2dv − nu2 , (65) for ballistic, collisionless processes in nanometric solid- state devices, even at relatively low densities n0 ∼ we get 1024m−3. The correspoinding Fermi temperature T ∼ F ∂ n ∂(nu) 40K, much smaller than the typical room tempera- + = 0, (66) ∂ t ∂ x ture T ∼ 300K, justifies the non-degeneracy assump- ∂ u ∂ u e ∂φ 1 ∂ P tion and Maxwell-Boltzmann’s statistics. + u = − . (67) ∂ t ∂ x m ∂ x mn ∂ x A more detailed theory would include the energy trans- 4 Derivation of a Fluid Model for Quantum Plasmas port equation obtained after taking the second-order moment of the Wigner function and the associated The Wigner-Poisson method presents some drawbacks: time-derivative. (a) it is a nonlocal, integro-differential system; (b) its Equations (66)and(67) coincide with the ordinary numerical treatment requires the discretization of the evolution equations for a classical fluid. This may seem entire phase space. Moreover, as is often the case with strange, but in the following it will be seen that the kinetic models, the Wigner-Poisson system gives more quantum nature of the system is in fact hidden in the information than one is really interested in. pressure term. Contributions explicitly dependent on For these reasons, it would be useful to obtain an are only found in the higher-order moments. For accurate reduced model which, though not providing this reason, since in the electrostatic case appears the same detailed information as the kinetic Wigner- explicitly only in the equation of motion for the third- Poisson approach, might still be able to reproduce the order moment [38], it is not enough to take only the main characteristics of quantum plasmas. energy transport equation into account. To obtain a set of macroscopic equations for quan- To proceed, first notice the equivalence between tum plasmas, we first derive a system of reduced ‘fluid’ the Wigner-Poisson and a system of countably many equations by taking moments of the Wigner-Poisson Schrödinger equations coupled to the Poisson equation, system. Using a Madelung (or eikonal) decomposition, one that has been mathematically demonstrated [39]. 358 Braz J Phys (2011) 41:349–363 More explicitily, the reduced one-body Wigner func- To be inline with the mean field approximation, it is tion can always be written as natural to factorize the N-body wavefunction in the form Nm M f(x,v,t) = pα ds ψ N ( , , ..., , ) = ψ ( , ) × ... × ψ ( , ), 2 π α x1 x2 xN t α x1 t α xN t (72) α=1 imv s ∗ s s × exp ψα x + , t ψα x − , t , which fully neglects correlations. Quantum statistics 2 2 effects are not taken into account in the Ansatz (72), (68) which does not respect the Pauli principle. With this caveat,weinsert(72)into(71), which yields precisely ≥ (68), with the same statistical weights pα. Hence we can with an ensemble probability pα 0, subject to the M = view the one-body ensemble wavefunctions ψα(x, t) as condition α=1 pα 1, for each one-particle ensem- ble wavefunctions ψα(x, t) normalized to unity and the result of splitting the N-body ensemble wavefunc- satisfying tion into the product of identical factors. Of course, it is hardly surprising that a correlationless model can 2 2 ∂ψα ∂ ψα ultimately be written in terms of a collection of one- i =− − eφψα,α= 1, ..., M, (69) ∂t 2m ∂x2 body Schrödinger equations. Thanks to the Schrödinger-Poisson form, we are able which is the Schrödinger equation for a particle un- to decompose the pressure term in a classical and a der the action of the mean field electrostatic potential quantum part, as follows. Consider the Wigner distri- φ(x, t). In addition, the Poisson (55) is rewritten as bution in (68). In terms of the ensemble wavefunctions, from (65) one obtains ∂2φ e M 2 M = N pα|ψα(x, t)| − n0 . (70) ∂x2 ε = |ψ |2, 0 α=1 n N pα α (73) α=1 Equations (69)and(70) constitute the so-called M ∗ iN ∂ψα ∗ ∂ψα Schrödinger-Poisson system, which has to be supple- nu = pα ψα − ψα , (74) 2m ∂x ∂x mented with suitable initial and boundary conditions. It α=1 provides a way of replacing the original N-body prob- lem by a collection of one-body Schrödinger equations, and, after some work, coupled by Poisson’s equation. From a methodological 2 M ∂ψ 2 ∂2ψ ∂2ψ∗ point of view, the Schrödinger-Poisson modeling cor- N α ∗ α α P = pα 2 − ψ − ψα responds to putting the emphasis again on the wave- ∂ α ∂ 2 ∂ 2 4m α= x x x function and not on the (phase-space) Wigner function. 1 2 Collective effects are mediated by the self-consistent 2 2 M ∗ N ∗ ∂ψα ∂ψα potential φ. + pα ψα − ψα . (75) 4mn ∂x ∂x A rigorous proof of the equivalence between (69) α=1 and (70) and the Wigner-Poisson system [39] is beyond the scope of the present text. We can nonetheless gain Now we follow Madelung [40] to decompose the some insight on the interpretation of the ensemble wavefunctions in the form wavefunctions. From (31)and(33), we see that ψ ( , ) = ( , ) ( , )/ , α x t Aα x t exp iSα x t (76) N f(x1,v1, t) = dx2dv2...dxNdvN f (x1,v1, ..., xN,vN, t) where the amplitudes Aα and phases Sα are real func- tions. We then find the expressions m M = N pα ds1 dx2...dxN 2 π M α=1 = 2 , v n N pα Aα (77) im 1s1 N ∗ s1 α= × exp ψα x + , x , ..., x , t 1 1 2 2 N M N ∂ Sα s1 2 × ψ N − , , ..., , . nu = pα Aα , (78) α x1 x2 xN t (71) m ∂x 2 α=1 Braz J Phys (2011) 41:349–363 359 and also ities. Consider the following redefined average < fα > of an ensemble function fα = fα(x, t): 2 M ∂ ∂ 2 N 2 2 Sα Sβ P = pα pβ Aα Aβ − M 2mn ∂x ∂x α,β=1 < fα >= p˜α fα. (87) α=1 2 M 2 2 N ∂ Aα ∂ Aα + pα − Aα . (79) Equations (82)and(83) are equivalent to the standard 2m ∂ x ∂ x2 α=1 deviations k 2 2 The constant now appears explicitly on the right-hand P = mn( uα− uα ), (88) side of the expression for the pressure. Notice, on the o = ( [ o]2− o2). other hand, that since the ensemble wavefunctions sat- P mn uα uα (89) isfy the one-body Schrödinger (69) both the amplitudes For a pure state so that p˜α = δαβ for some β, both Pk and phases implicitly depend on Planck’s constant. and Po vanish, and only PQ survives. It is useful to define the kinetic velocity uα and the Since the kinetic and osmotic pressures measure the o osmotic velocity uα associated to the wavefunction ψα: kinetic and osmotic velocity dispersions, it is reason- ∂ ∂ /∂ able, even if not rigorous, to assume an equation of 1 Sα o Aα x uα = , and uα = . (80) state so that m ∂x m Aα Pk + Po = PC(n), (90) Then it can be verified that the pressure in (79)is given by dependent only on the density. In this way, we obtain 2 2 k o Q n ∂ P = P + P + P , (81) P = PC(n) − ln n. (91) 4m ∂x2 k where the kinetic pressure P is For definiteness, we call PC the “classical” part of the M pressure, in the sense that it measures the dispersion of k mn 2 P = p˜α p˜β (uα − uβ ) , (82) the velocities. However, it explicitly contains Planck’s 2 α,β=1 constant since it depends on the osmotic velocities, which always have a purely quantum nature. the osmotic pressure Po is The replacement of the sum of the kinetic and os- motic pressures by a function of the density requires M o mn o o 2 only a few comments. Although exact, (81) offers no P = p˜α p˜β (uα − uβ ) , (83) 2 α,β=1 advantage over the Wigner-Poisson formulation, be- cause it ultimately requires knowledge of the ensemble and the quantum pressure PQ is wavefunctions, and then the solution of a countable set of self-consistent Schrödinger equations. In classical 2n ∂2 PQ =− ln n. (84) kinetic theory (i.e., without the osmotic and quantum 4m ∂x2 pressures), it is customary to reach closure by assuming In (82)and(83), a modified set of ensemble probabili- that the standard deviation of the velocities is a function of the density only. We have gone just one step further, ties p˜α = p˜α(x, t) was introduced, including also the standard deviation of the osmotic 2 Npα Aα velocities. p˜α = . (85) n In addition, in the classical limit we expect equa- tions reproducing the classical fluid equations. This is M ˜ ≥ , ˜ = k o C The new statistical weights satisfy pα 0 α=1 pα 1, certainly true if P + P = P (n) for some appropriate as they should. function of density only. Finally, the standard Euler For a particular ψα the osmotic velocity points to the equations are reproduced thanks to the residual clas- regions of higher density [41, 42], as is more evident in sical limit in Pk. Alternatively, for ≡ 0,wealsohave the three-dimensional version, Po = 0 and we expect Pk to be some function of the density only. o uα = ∇ ln Aα. (86) Unlike P in (65), which uses the Wigner function, m the statistical weights in (88)and(89) are provided Both pressures Pk and Po can be viewed as a mea- by the p˜α in (85). For a pure state one has PC = 0, sure of the dispersion of the kinetic and osmotic veloc- which is inline with the understanding that a pure state 360 Braz J Phys (2011) 41:349–363 corresponds to a cold plasma with no dispersion of With the hypothesis (90), the force equation (67)can velocities. The contribution PQ, on the other hand, be written as finds no classical counterpart at all. ∂ u ∂ u 1 ∂ PC(n) e ∂φ 1 ∂ PQ It has long been known that closure poses a delicate + u =− + − . (94) ∂ t ∂ x mn ∂ x m ∂ x mn ∂ x problem. The derivation of macroscopic models from microscopic models always call for some degree of Using the identity √ approximation and a more or less phenomenological 1 ∂ PQ 2 ∂ ∂2( n)/∂ x2 or ingenuous point of view. The present approach is =− √ , (95) mn ∂ x 2m2 ∂x n capable of taking into account the quantum statistics of the charge carriers, represented by an appropriated we can rewrite the basic quantum hydrodynamic model equation of state. For example, assuming interaction for plasmas in terms of the continuity (66) and the force with a heat bath an isothermal equation of state is indi- equation cated. For fast phenomena, in which thermal relaxation ∂ u ∂ u 1 ∂ PC(n) e ∂φ 2 ∂ has no time to occur, an adiabatic equation of state can + u =− + + ∂ t ∂ x mn ∂ x m ∂ x 2m2 ∂x be used. √ ∂2( n)/∂ x2 In the dilute or dense classes of plasmas, Maxwell- × √ . (96) Boltzmann or Fermi-Dirac distributions can be em- n ployed, respectively. Moreover, the model accounts for In the limit → 0 this is formally equal to Euler’s equa- quantum diffraction effects—in particular for tunneling tion for an electron fluid in the presence of an electric and wave packet dispersion—, which are present in the field −∂φ/∂x. Finally, we have the Poisson equation quantum part PQ of the pressure. It is also able to ∂2φ reproduce the linear dispersion relation from kinetic = e ( − ), 2 n n0 (97) theory if the equation of state is adequate, except for ∂ x ε0 purely kinetic phenomena. Finally, the quantum fluid The only difference from classical plasmas is the ∼ 2 model reduces to the standard Euler equations in the term in (96), the so-called Bohm potential term. While formal classical limit and are sufficiently simple to be mathematically the Bohm potential is equivalent to a amenable to efficient numerical simulation. pressure to be inserted in the momentum transport The proposed simplification becomes more justified equation, physically it corresponds to typical quantum for an important class of statistical ensembles, in which phenomena like tunneling and wave packet spreading. all the wavefunctions have equal amplitudes, although Therefore it is not a pressure in the thermodynamic not necessarily constant ones, sense. Besides, it survives even for a one-particle pure- state system. n iSα / ψα = e . (92) Many quantum hydrodynamical models are popular N in the context of semiconductor physics. For instance, Gardner considered a quantum-corrected displaced Then the osmotic pressure identically vanishes and one Maxwellian distribution [43], as introduced by Wigner has p˜α = pα,sothat(88) becomes the usual standard [19]. More exactly, it is possible to find the leading deviation of the kinetic velocities. Hence Pk can be quantum correction f (x,v,t) for a momentum-shifted interpreted in full analogy with the standard thermody- 1 local Maxwell-Boltzmann equilibrium f (x,v,t),set- namic pressure. The velocity dispersion stems directly 0 ting f(x,v,t) = f (x,v,t) + 2 f (x,v,t) in the semi- from the randomness of the phases of the wavefunc- 0 1 classical quantum Vlasov (56) and collecting equal tions. The approximation can be viewed as a first step powers of 2. In other words set beyond the standard homogeneous equilibrium of a / fermion gas, for which each state can be represented m 1 2 f0(x,v,t) = n(x, t) by a plane wave 2πκBT(x, t) −m[v − u(x, t)]2 ψα(x, t) = A0 exp (imuα x/), (93) × exp , (98) 2κBT(x, t) with spatially constant amplitude A0 and velocities uα. on the assumption of a non-degenerate quasi- In the generalization (92), both the amplitude and the equilibrium state. After calculating f1,insert 2 velocity can be spatially modulated, although the am- f = f0 + f1 in the pressure in (65). A quantum plitude is the same for all states. For systems not too far hydrodynamical model similar to (66), (96), and (97) from equilibrium, this seems reasonable. is then found (and generalized by the inclusion of Braz J Phys (2011) 41:349–363 361 an energy transport equation). By definition, the The gradient dependence of the stress tensor is due resulting system is restricted to quasi-Maxwellian, to quantum mechanical nonlocality. Once again, the dilute systems. We also observe that in the case of proof in [45] is valid when quantum contributions to a self-consistent problem the electrostatic potential the self-consistent mean field potential can be ignored: should have been also expanded in powers of some no expansion of φ in powers of a quantum parameter non-dimensional quantum parameter, which was not was used. Moreover, the plasma is assumed to be dilute done in [43]. and in the semiclassical regime, in which Maxwell- The conclusion is that the procedures involving a Boltzmann statistics is applicable and a small dimen- Madelung decomposition of the quantum ensemble sionless quantum parameter exists. wavefunctions or a quantum corrected Wigner function Further, quantum hydrodynamic models with a equilibrium involve working hypotheses which are not “smoothed” potential have been derived, to handle rigorously justified. Starting from the Wigner-Poisson discontinuities in potential barriers in semiconductors system, which is by definition collisionless, and relying [46]. Energy transport [47] and quantum drift-diffusion on quasi-equilibrium assumptions, one could hardly [48] have been also employed to model transport in derive rigorous macroscopic theories. Nevertheless, the ultra-small electronic devices, including the addition of numerical and analytical advantages of quantum fluid viscosity effects [49]. models over quantum kinetic models make them much Quantum hydrodynamic models are well established more popular than the kinetic treatment, to approach in several specialties, even if the individual communi- the nonlinear aspects of quantum plasma physics. ties are unaware of the widespread dissemination. In Alternatively, general thermodynamic arguments by molecular physics, for instance, the simplest gradient Ancona and Tiersten have been used to derive the functional theory is given by the Thomas-Fermi-Dirac- Bohm potential. Ancona and Tiersten [44] argues that von Weizsäcker functional, which provides a suitable the internal energy of the electron fluid in an electron- expression for the energy density of molecules [50]. Its hole semiconductor should depend not only on the constituents are a Thomas-Fermi term for the uniform density, but also on the density gradient, in order to ex- electron gas [51, 52], a Dirac exchange functional [53], tend the standard drift-diffusion model so as to include the Coulomb term for the electron gas, a nuclear at- the quantum-mechanical behavior in strong inversion traction term, and the von Weizsäcker term [54], with layers. Their method defines a “double-force” and a an adjustable fitting parameter. With this phenomeno- “double-pressure vector” which allows for changes of logical parameter set to unity, the von Weissäcker term the internal energy of the electron gas purely due to gives exactly the Bohm potential. density fluctuations. It is useful to remember the obvious limitations of By postulating a linear dependence of the double- the quantum hydrodynamic model for plasmas: (a) pressure vector on density gradients (see (3.3) of [44]) since it is a fluid model, it is applicable to long wave- and working out the conservation laws of charge, mass, lengths only. This condition is expressed roughly as linear momentum and energy, Ancona and Tiersten λ>λD for non-degenerate and λ>λF for degenerate found a generalized chemical potential comprising two systems, where λ is the pertinent wavelength. Kinetic contributions: (a) a gradient-independent term which phenomena, such as Landau damping or the plasma can be modeled by the equation of state of a zero- echo [55], call for detailed knowledge of the equilib- temperature Fermi gas or any other appropriated form. rium Wigner function and hence require kinetic treat- This corresponds to the classical pressure PC(n) of ment; (b) no energy transport equation was included. the quantum hydrodynamical model for plasmas; (b) a This limitation, however, is not definitive, since it is a Bohm potential term proportional to a phenomenolog- simple exercise to obtain the equation from the second- ical parameter initially left free. order moment of the quantum Vlasov equation; (c) it Later, Ancona and Iafrate [45] obtained the expres- applies only to non-relativistic phenomena; (d) no spin sion of the phenomenological coefficient in [44]. In effects are included, except for the equation of state [45], which is similar to [43], the first order quantum which can, to a certain extent, represent some quantum correction for a Maxwell-Boltzmann equilibrium was statistical effects. It suffices to apply the equation of found from the semiclassical quantum Vlasov equation state for a dense, degenerate electron gas. In the same and then employed to calculate the particle density and way, some relativistic effects can be also incorporated the stress tensor. The potential function eliminated be- with an equation of state for a relativistic electron gas; tween the two expressions, an equation of state relating (e) no magnetic fields are in the present formulation; (f) the stress tensor and the particle density and gradient the sum of kinetic and osmotic pressures is assumed to was derived, containing the Bohm potential. be a function of the density only. This is in the spirit of 362 Braz J Phys (2011) 41:349–363 density-functional theories and appears reasonable in tion the Wigner-Fokker-Planck system approach to the vicinity of homogeneous equilibria, but may prove model quantum dissipation [34, 71]. inappropriate far from equilibrium; (g) strongly cou- pled plasmas are out of reach since the starting point Last but not least, we mention that this introductory for the derivation was the quantum Vlasov equation, text was focused on the detailed derivation of the basic valid only for weak correlations. kinetic and hydrodynamic models for quantum plas- It is remarkable that the eikonal decomposition mas, in a language meant to be accessible to beginners. method, starting from the Wigner-Maxwell system, al- A more complete account of the current applications of lows inclusion of magnetic fields [37], the resulting such quantum plasma models can be found in [1–4]. quantum hydrodynamic model being modified simply by the addition of a magnetic ∼u × B force. 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