One-Particle Density Matrix for a Quantum Gas in the Box Geometry
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug One-Particle Density Matrix for a Quantum Gas in the Box Geometry
Soumi Dey, Srijit Basu, Debshikha Banerjee, and Shyamal Biswas ∗ School of Physics, University of Hyderabad, C.R. Rao Road, Gachibowli, Hyderabad-500046, India (Dated: August 7, 2019)We have analytically obtained 1-particle density matrices for ideal Bose and Fermi gases in 3-D boxgeometries for the entire range of temperature. We also have obtained quantum cluster expansionsof the grand free energies in closed forms for the same systems in the restricted geometries. We alsohave considered short ranged interactions in our analyses for the quasi 1-D cases of Bose and Fermigases in the box geometries. Our results are exact and are directly useful for a postgraduate courseon statistical mechanics or that on many-body physics.
PACS numbers: 01.40.Ha Learning Theory and Science Teaching, 05.30.-d Quantum Statistical Mechanics,05.30.Fk Fermion Systems and Electron Gas, 05.30.Jp Boson Systems
I. INTRODUCTION
Density matrix is of very high interest in physics [1–4]. It maps equilibrium statistical mechanics to quantumdynamics and vice versa with the application of Wickrotation (1 /k B T ⇆ it/ ~ ) which maps inverse temperate(1 /T ) to imaginary time ( it ) [4, 5]. Density matrix ele-ments, which physically represents spatial correlations ina thermodynamic or mechanical system, are nothing butthe propagators in the position representation [4]. Den-sity matrix elements are thus useful to get path integralsin statistical field theory and quantum field theory [4].Density matrix is often introduced in several postgrad-uate courses on physics such as statistical mechanics,many-body physics, quantum mechanics, quantum fieldtheory, quantum optics, etc [4, 6]. Density matrix ele-ments (and 1-particle density matrix elements for many-body systems [7, 8]) are often calculated in position rep-resentation in the class for systems having no boundariesat all, e.g. a free particle, a harmonic oscillator, a freeBose gas at a temperature T , a free Fermi gas at a tem-perature T , etc [4, 9]. There have been many discussionson in the frontiers level in connection with the 1-particledensity matrices or cluster expansions or spatial correla-tions in quantum gases [10–14]. However, hardly any dis-cussions are found on the finite size effects on the densitymatrices in the graduate-course text-books or even in theresearch articles. Practically all the thermodynamic sys-tems which come to equilibrium with the respective heat(and in some occasions particle) reservoirs are bounded.Hence we are interested in calculating 1-particle densitymatrix [8] for ideal Bose and Fermi gases in 3-D box ge-ometries at a temperature T . We are also interested toexplore the same interacting Bose and Fermi gases con-fined at least in quasi 1-D boxes. We also want to havequantum cluster expansions [15, 16] of the grand free en-ergies for these systems in connection with the density ∗ Electronic address: sbsp [at] uohyd.ac.in matrices. Thermodynamic properties of the systems andthe finite-size effects on them can be easily obtained fromthe quantum cluster expansions of the grand free ener-gies.Study of ultracold quantum gases has been a topic ofhigh experimental and theoretical interest [17–19] afterthe observation of Bose-Einstein condensation of alkaliatoms in 3-D magneto-optical harmonic traps in 1995[20–22]. Bose-Einstein condensation of a dilute Bose gasof Rb atoms has also been observed recently in a 3-Dmagneto-optical box trap with quasi-uniform potential init [23]. Thus studying 1-particle density matrix for idealBose and Fermi gases in 3-D box geometries and thatof quantum cluster expansions of these systems wouldbe relevant in the current context not only from science-education point of view but also from research point ofview.Calculations of this article begin with the introduc-tion of the statistical mechanical density matrix and theequation for its matrix-element in position-space repre-sentation for a single particle in equilibrium with a heatbath at an absolute temperature T . Then solve the equa-tion for the density matrix-element for a particle in 1-Dbox. Then we generalize the solution for 3-D box. Thenwe generalize the 3-D case of the single particle to theideal gases of many-particles (i.e. for identical bosons andfermions) with 1-particle density matrix within grand-canonical ensemble. Then we obtain quantum cluster ex-pansions for grand free energies for these systems. Thenwe consider short ranged interactions in our analyses forthe quasi 1-D cases of Bose and Fermi gases in the boxgeometries. We conclude finally. II. STATISTICAL MECHANICAL DENSITYMATRIX
Let us consider a thermodynamic system in equilib-rium with a heat bath at an absolute temperature T .Statistical mechanical density matrix is defined for the / L0.20.40.60.81.0 (cid:1) T ρ ( x,x'; β ) For λ T / L = (cid:0)
3; x (cid:2) L = (cid:3) ( solid line ) , 1 / ( dashed line ) FIG. 1: Profile of the density matrix elements for the particlein the 1-D box of length L . The solid and dashed lines followEqn. (8) for x/L = 1 / / x/L = 1 /
2. Thermal de Broglie wavelength of the system, λ T = q π ~ βm , is set as λ T = 2 L/ system as [4] ˆ ρ = e − β ˆ H (1)where ˆ H is the Hamiltonian of the system, β = 1 /k B T and k B is the Boltzmann constant. Though a normal-ization factor 1 / Tr . ˆ ρ is needed to normalize the densitymatrix, we are not considering it in the definition. Inverseof the normalization factor is called the partition func-tion ( Z = Tr . ˆ ρ [4]), which can be separately consideredwhenever needed to extract thermodynamic propertiesof the system. If {| ψ j i} ; j = 1 , , , .... be the orthonor-malized ( h ψ i | ψ j i = δ i,j ∀ i, j ) and complete set of en-ergy eigenstates ( P ∞ j =1 | ψ j ih ψ j | = ) of the system withthe respective eigenvalues { E j } , then density matrix inEqn.(1) can be recast as [4]ˆ ρ = ∞ X j =1 e − βE j | ψ j ih ψ j | (2) A. Density matrix for a single particle in a 1-D box
If the points x and x ′ be any two arbitrary points insidea 1-D box of length L which confines a point-particle suchthat 0 < x < L , then the density matrix element of thesystem can be defined in position representation as ρ ( x, x ′ ; β ) = h x | ˆ ρ | x ′ i = h x | e − β ˆ H | x ′ i = ∞ X j =1 e − βE j ψ j ( x ) ψ ∗ j ( x ′ ) (3) The definition leads to the equation for the densitymatrix-element as ∂∂β ρ ( x, x ′ ; β ) = − ˆ H x ρ ( x, x ′ ; β ) = ~ m ∂ ∂x ρ ( x, x ′ ; β ) , (4)where ˆ H x = − ~ m ∂ ∂x is the position representation of theHamiltonian of the system in the nonrelativistic domain,and m is the mass of the system. By sending β → ρ ( x, x ′ ; 0) = δ ( x − x ′ ). This can beconsidered as an inhomogeneous boundary condition forthe solution to the diffusion Eqn. (4). The solution for L → ∞ , i.e. for the case of free particle, ρ f ( x, x ′ ; β ) = r m π ~ β e − m ( x ′− x )22 ~ β (5)is well known in the literature [4].The solution to the Eqn.(4) for finite L , however, isnot known in the literature in compact form. Dirichletboundary conditions on the energy eigenstates in Eqn.(3)leads to take the form of the solution, for finite L , as ρ ( x, x ′ ; β ) = ∞ X j =1 ˜ ρ ( j, x ′ ; β ) sin (cid:0) jπxL (cid:1) (6)where sin (cid:0) jπxL (cid:1) = q L ψ j ( x ) = q L h x | ψ j i is q L times the normalized energy eigenstate for the par-ticle in the box, and the Fourier series expansion-coefficient, ˜ ρ ( j, x ′ ; β ), is to be determined from Eqn.(4)by performing the inverse transformation for thesame inhomogeneous boundary condition. Thus,the expansion-coefficient takes the form ˜ ρ ( j, x ′ ; β ) = q L sin (cid:0) jπx ′ L (cid:1) e − βj π ~ mL so that the solution becomes ρ ( x, x ′ ; β ) = ∞ X j =1 L e − β j π ~ mL sin (cid:0) jπx ′ L (cid:1) sin (cid:0) jπxL (cid:1) = ∞ X j =1 L e − β j π ~ mL (cid:20) cos (cid:18) jπ [ x ′ − x ] L (cid:19) − cos (cid:18) jπ [ x ′ + x ] L (cid:19)(cid:21) (7)This form of solution also directly follows from Eqn.(3)as the energy eigenvalues are E j = π ~ j mL for ∀ j for theparticle in the 1-D box. However, by performing thesummation over j in Eqn.(7) we get an exact result incompact form as ρ ( x, x ′ ; β ) = 12 L (cid:20) ϑ (cid:18) π [ x ′ − x ]2 L , e − β π ~ mL (cid:19) − ϑ (cid:18) π [ x ′ + x ]2 L , e − β π ~ mL (cid:19)(cid:21) (8)where ϑ represents a Jacobi (elliptic) theta function inthe usual notation[31]. The second term in the right handside of Eqn.(8) breaks the translational symmetry in thedensity matrix-element, as because, the density matrix-element no longer depends on separation of the points x and x ′ for finite L . We plot the density matrix-element(Eqn.(8)) and also compare it with the one with the free-boundary (Eqn.(5)) in FIG. 1. Eqn.(8) is significantlydifferent from Eqn.(5) in the low temperature regimewhen thermal de Broglie wavelength, λ T = q π ~ mk B T , be-comes comparable to or bigger than the system size ( L ).It is also clear from the dashed line of the figure, that,the density matrix-element is not being maximized at x ′ = x unless x = L/ L ) of the system. B. Density matrix for a single particle in a 3-D box
Let us now consider the particle to be in a 3-D boxof lengths L , L , L along the x, y, z axes respectively.Position of the particle is now given by ~r = x ˆ i + y ˆ j + z ˆ k (or ~r ′ = x ′ ˆ i + y ′ ˆ j + z ′ ˆ k ) such that 0 < x < L , 0 < y < L ,0 < z < L . Since the system is linear, as there are nointer particle interactions, motions of the particle along x , y and z axes would be independent of each other.Thus, we can generalize the result for the density matrix-element in Eqn.(8) for the 3-D as product of the densitymatrix elements for the individual axes: ρ ( ~r, ~r ′ ; β ) = Π i =1 L i (cid:20) ϑ (cid:18) π [ x ′ i − x i ]2 L i , e − β π ~ mL i (cid:19) − ϑ (cid:18) π [ x ′ i + x i ]2 L i , e − β π ~ mL i (cid:19)(cid:21) (9)where x = x , x = y , x = y , and so as for the primedcoordinates.Eqn.(9) is our result for statistical mechanical densitymatrix element ( ρ ( ~r, ~r ′ ; β ) = h ~r | ˆ ρ | ~r ′ i ) for a single parti-cle in a 3-D box in equilibrium with a heat bath at atemperature T . In the following section, we extend thestatistical mechanical density matrix for many-body sys-tems within grandcanonical ensemble. III. ONE-PARTICLE DENSITY MATRIX FORAN IDEAL QUANTUM GAS IN A 3-D BOX
One-particle density matrix for an ideal quantum gas( i.e. Bose or Fermi gas of indistinguishable particles) inequilibrium with a heat (and particle) bath at a temper-ature T (and chemical potential µ ) is defined (by gener-alizing Eqn.(2)) as [7, 8]ˆ ρ = ∞ X j =1 β ( E j − µ ) ± | ψ j ih ψ j | (10) where {| ψ j i} are the orthonormalized and complete setof eigenstates of eigenvalues {| E j i} for the single-particleHamiltonian of the many-body system and the pre-factor, β ( Ej − µ ) , represents the average occupation num-ber of particles (¯ n j ) to the single-particle state | ψ j i . Thuswe generalize Eqn.(9)to the 1-particle density matrix forthe many-body system in the 3-D box as ρ ( ~r, ~r ′ ; β ) = ∞ X j =1 ∞ X j =1 ∞ X j =1 β ([ E j + E j + E j ] − µ ) ∓ × Π i =1 L i sin (cid:18) j i πx i L i (cid:19) sin (cid:18) j i πx ′ i L i (cid:19) (11)where E j i = π ~ j i mL i for i = 1 , ,
3, and upper sign repre-sents Bose gas and lower sign represents Fermi gas.While single-particle ground state energy of the sys-tem is given by ˜ E = π ~ m (cid:2) L + L + L (cid:3) , fugacityof the system is given by ¯ z = e µ/k B T which ranges as0 ≤ ¯ z < e ˜ E /k B T for the ideal Bose gas and 0 ≤ ¯ z < ∞ for the ideal Fermi gas. One-particle density matrix-element (in position representation) catches long-rangedorder in the many-body system at least for 3-D free Bosegas [7, 8]. The finite size effect kills the long-ranged orderin the many-body system as the density matrix elementhas to vanish at the boundaries as clear even in FIG. 1.Expansion of the r.h.s. of Eqn.(11) around ¯ z = 0 leadsto ρ ( ~r, ~r ′ ; β ) = ∞ X l =1 ( ± l − ¯ z l ρ ( ~r, ~r ′ ; lβ ) . (12)If boundaries along x and y axes are removed, i.e. if L and L are sent to ∞ , then the system would be confinedonly along the z -axis. The one-particle density matrix inthis situation would be ρ ( ~r, ~r ′ ; β ) = ∞ X l =1 ( ± l − ¯ z l l ρ f ( ~r ⊥ , ~r ′⊥ ; lβ ) ρ ( z, z ′ ; lβ ) (13)where ~r ⊥ = x ˆ i + y ˆ j , ~r ′⊥ = x ′ ˆ i + y ′ ˆ j , ρ f ( ~r ⊥ , ~r ′⊥ ; β ) = 1 λ T e − π | ~r ⊥− ~r ′⊥| λ T (14)is the density matrix for a free-particle in the x − y plane[4], and ρ ( z, z ′ ; β ) is the density-matrix of the particle,similar to that in Eqn.(8), for the bounded motion alongthe z -axis. For 3-D free quantum gas, Eqn.(13) takes theform [7, 8] ρ f ( ~r, ~r ′ ; β ) = 1 λ T ∞ X l =1 ( ± l − ¯ z l l / e − π | ~r − ~r ′| lλ T (15)The term with the first power of the fugacity ¯ z in the den-sity matrix-element in Eqn.(13) corresponds to the clas-sical case. We plot the 1-particle density matrix-element z' / L (cid:17) T ρ ( r , r '; β ) λ T / L = /
3, z / L = /
2, x = x', y = y', and z = (cid:18)(cid:19)(cid:20) for Bose ( solid line ) , Fermi ( dashed line ) , and free Bose ( dotted line ) gases FIG. 2: Profile of the 1-particle density matrix-elements forthe 3-D ideal Bose gas (solid line), Fermi gas (dashed line),which are free in x − y plane and confined in the 1-D boxof length L along z -axis, for the fugacity ¯ z = 0 .
8. Solidand dashed lines follow Eqn.(13) for upper and lower signsrespectively. The dotted line represents the same for 3-D freeBose gas, and follows Eqn.(15) for the upper sign. in Eqn.(8) for 3-D ideal Bose gas, and compare it with theone (in Eqn.(15)) with the free-boundary in FIG. 2. It isclear from the FIG. 2 that the box-confinement reducesthe spatial correlations in the system in comparison tothat in the free quantum (Bose or Fermi) gas. The finitesize effect kills the long-ranged order in the 3-D idealBose gas as the density matrix element has to vanish atthe boundaries as clear in the FIG. 2. The dotted linewould come arbitrary close to the solid line at z = L / z → ρ ( ~r, ~r ′ ; β ) = ∞ X j =1 (cid:20) Z e i ( ~r ′⊥ − ~r ⊥ ) · ~p ⊥ / ~ e β ([ p ⊥ / m + E j ] − µ ) + 1 d ~p ⊥ (2 π ~ ) × L sin (cid:18) j πzL (cid:19) sin (cid:18) j πz ′ L (cid:19)(cid:21) = ∞ X j =1 (cid:20) Z ∞ πJ ( | ~r ′⊥ − ~r ⊥ | p ⊥ / ~ )e β ([ p ⊥ / m + E j ] − µ ) + 1 p ⊥ d p ⊥ (2 π ~ ) × L sin (cid:18) j πzL (cid:19) sin (cid:18) j πz ′ L (cid:19)(cid:21) , (16)which can be further recast for T →
0, for which µ ≥ π ~ mL and p ⊥ ranges from 0 to p F j = FIG. 3: Profile of the 1-particle density matrix-elements forthe 3-D ideal Fermi gas for T →
0. The 3-D plot followsEqn.(17). p mµ − π ~ j /L , as ρ ( ~r, ~r ′ ; ∞ ) = 1 πL ⌊ r mL µπ ~ ⌋ X j =1 (cid:20) sin (cid:18) j πzL (cid:19) sin (cid:18) j πz ′ L (cid:19) × p F j ~ J ( | ~r ′⊥ − ~r ⊥ | p F j / ~ ) | ~r ′⊥ − ~r ⊥ | (cid:21) . (17)We plot r.h.s. of Eqn.(17) in FIG. 3. The oscillations inthe density matrix-element in Eqn.(17) are coming fromthe alternation of the sign of ¯ z l in Eqn.(13). Amplitudeof the partial oscillations along x and y axes, however,is dying out hyperbolically as the system is not boundedalong these two axes. The amplitude of the partial os-cillations along the z -axis, however, are oscillating, asexpected, as the system is bounded along this axis. Theoscillatory amplitude oscillates rapidly except at around z = z ′ if the Fermi energy (i.e. µ for T →
0) of the systemincreases; which further causes increase of the principalmaximum of the density matrix-element of the system.Spatial density correlation in the many-body systemcan represented in terms of the 1-particle density ma-trix as ν ( ~r, ~r ′ ; β ) = ± | ρ ( ~r,~r ′ ; β ) | ρ ( ~r,~r ; β ) [24]. All the results wehave got in this sections are thus useful to get the spacialdensity correlations in the many-body systems. Theseresults can also be useful to get the quantum cluster ex-pansions for the grand free energies of the many-bodysystems confined in the box geometries. IV. QUANTUM CLUSTER EXPANSION FORAN IDEAL QUANTUM GAS IN 3-D BOX
Quantum cluster expansion of the grand free energyfor a 3-D ideal quantum gas is given by [15, 16]Ω = − k B T ∞ X ν =1 ( ± ν − h ν ¯ z ν ν (18)where h ν = Z .. Z ρ ( ~r , ~r ; β ) ρ ( ~r , ~r ; β ) ...ρ ( ~r ν , ~r ; β )d ~r ... d ~r ν (19)is the cluster integral for ν indistinguishable particles inthe system and ~r i ( i = 1 , , ..., ν ) is the position vectorfor a particle in the cluster, and ρ ( ~r i , ~r j ; β ) is the single-particle density matrix element for the two positions ~r i and ~r j as defined in Eqn.(9). Eqn.(9) leads to take thesimplest form for the cluster integral h as h = Z L Z L Z L Π i =1 L i (cid:20) ϑ (cid:18) π [ x i − x i ]2 L i , e − β π ~ mL i (cid:19) − ϑ (cid:18) π [ x i + x i ]2 L i , e − β π ~ mL i (cid:19)(cid:21) d x d x d x = ∞ , ∞ , ∞ X j ,j ,j =1 e − β π ~ m (cid:0) j L + j L + j L (cid:1) . (20)Fourier series expansion of the Jacobi (elliptic) thetafunctions, as was as the Fourier series expansion shownin Eqn.(7) for the density matrix-element for a particlein 1-D box, has been used to get the final expression of h which by definition is the partition function for a sin-gle particle in the 3-D box. It can be shown from theorthonormality of the single-particle energy eigenstatesthat, two indistinguishable particles in the different en-ergy eigenstates do not contribute to the cluster integral.Thus, by the definition, cluster integral for the two orany number of indistinguishable particles would be thepartition function for the same number of particles al-ways in the same single-particle energy eigenstate. Thuswe get h ν = ∞ , ∞ , ∞ X j ,j ,j =1 ( ± ν − e − νβ π ~ m (cid:0) j L + j L + j L (cid:1) , (21)for ν = 1 , , , ... . Now we get the quantum cluster ex-pansion of the ideal quantum (Bose or Fas) gas in a 3-Dbox, by recasting Eqn.(18), asΩ = − k B T ∞ X ν =1 ( ± ν − ¯ z ν ν ∞ , ∞ , ∞ X j ,j ,j =1 e − νβ π ~ m (cid:0) j L + j L + j L (cid:1) . (22)Average number of particles ( ¯ N = − ∂ Ω ∂µ | T,L L L ), on theother hand can be calculated from Eqn.(22) as¯ N = ∞ X ν =1 ( ± ν − ¯ z ν ∞ , ∞ , ∞ X j ,j ,j =1 e − νβ π ~ m (cid:0) j L + j L + j L (cid:1) . (23) L / λ T βλ T p L →∞ , L →∞ , and z = ( solid line ) , Fermi ( dashed line ) , and free Bose ( dotted line ) gases FIG. 4: Length dependence of the equation of states for the3-D ideal Bose gas (solid line), Fermi gas (dashed line) andfree Bose gas for fixed fugacity. The 1st two plots followsEqn.(24) for +ve and -ve signs respectively.
Let us now define the generalized pressure given by thequantum gas on a wall of the system as p = − Ω L L L whichwould be the true pressure in the thermodynamic limit.Thermodynamic behaviour of the ideal quantum gas cannow be extracted from Eqns. (22) and (23).Let us now remove the boundaries along the x and y axes so that L → ∞ , L → ∞ , and the quantum gasremains bounded only along the z axis. In this situationthe summations in Eqns. (22) and (23) over j and j can be replaced by integrations from 0 to ∞ without anyerror by virtue of Poisson summation formula. Thus weget the equation of state of the system, i.e. the pressuregiven by the quantum gas to either of the walls situatedat z = 0 and z = L , as p = k B TL λ T ∞ X ν =1 ( ± ν − ¯ z ν ν ∞ X j =1 e − νβ π ~ j mL = k B TL λ T ∞ X ν =1 ( ± ν − ¯ z ν ν (cid:2) ϑ (cid:0) , e − νβ π ~ mL (cid:1) − (cid:3) . (24)This is quantum cluster expansion of the equation of statefor ideal Bose and Fermi gases in the box geometry. Aswe mentioned before, the upper sign corresponds to theideal Bose gas and the lower sign corresponds to the idealFermi gas. We plot the equation of state in Eqn. (24) forideal Bose and Fermi gases for fixed fugacity, and com-pare with the the case of the free Bose gas in FIG. 4.The solid line approaches the result λ T p/k B T = g / (¯ z ),where g / is the Bose integral [16], for the 3-D free Bosegas as L /λ T tends to ∞ . The dashed line approachesthe result λ T βp = f / (¯ z ), where f / is the Fermi inte-gral [16], for the 3-D free Fermi gas as L /λ T tends to ∞ . It is clear from the FIG. 4 that, finiteness of systemcauses less pressure given to the wall with respect to thatgiven by a free gas though finiteness causes more energyto the system which is not probabilistic favoured in ther-mal equilibrium. This pressure is exponentially small for λ T (cid:19) L as clear from the FIG. 4 too. The finite sys-tem effectively behaves like a 2-D system in such a lowtemperature regime. This pressure, however, is furtherreduced in Fermi gas for odd permutation effect in clus-ters of even sizes. Casimir-like effect can also be studiedfrom Eqn. (24) for the finiteness of the system only alongthe z axis [25] apart from studying typical finite-size ef-fect on the many-body system. V. QUANTUM CLUSTER EXPANSION FOR ANON-IDEAL QUANTUM GAS IN A CLOSEDRECTANGULAR CYLINDER
Let the Bose gas spin 0 particles be interacting withshort-ranged pair-potential energy V ( | ~r − ~r ) | ) in a 3-D closed rectangular cylindrical box, so that the many-body Hamiltonian of the system can be written, in usualnotation, as [17]ˆ H = Z d ~r ˆ ψ † ( ~r ) (cid:18) − ~ m ∇ (cid:19) ˆ ψ ( ~r ) +12 Z Z d ~r ′ d ~r ˆ ψ † ( ~r ) ˆ ψ † ( ~r ′ ) V ( | ~r − ~r ′ | ) ˆ ψ ( ~r ′ ) ˆ ψ ( ~r )(25)where V ( | ~r − ~r ′ ) | ) = g δ ( | ~r − ~r ′ | )[33] is considered tobe a short-ranged pair-potentail energy, g = π ~ a s m isthe coupling constant, and a s is the s-wave scatteringlength which is positive for repulsive interactions. Thesystem would not be stable beyond a critical number ofparticles for g < L ≪ L and L ≪ L so that low-lying excitations only along z -direction are probabilistically favoured for low temper-atures (for which λ T ≫ L and λ T ≫ L ). The systembehaves like a quasi 1-D system (0 ≤ z ≤ L ) in this sit-uation. If the average number of identical bosons in thegrandcanonical ensemble (for temperature T and chem-ical potential µ ) be ¯ N , then the 1-particle (low-lying)excitations ( ψ j ( z ); j = 1 , , , ... ) follow from the time-independent non-linear Schrodinger equation [28] (cid:20) − ~ m d d z + g ¯ N | ψ j ( z ) | (cid:21) ψ j ( z ) = ¯ E j ψ j ( z ) (26)with the 1-particle mean field energy eigenvalue [28]¯ E j = 2 ~ j mL (1 + q j ) K ( q j ) , (27)where K ( q j ) = π (cid:2) q j / q j /
64 + ... (cid:3) is completeelliptic integral of the first kind of the argument q j = g ¯ N | ψ j ( L / j ) | E j / (cid:2) − g ¯ N | ψ j ( L / j ) | E j (cid:3) (for 0 < q j < / g = ~ ma s is the coupling constant for elastic scat-tering in 1-D, and [28] ψ j ( z ) = s ~ q j mL g ¯ N (cid:2) jK ( q j ) (cid:3) sn (cid:18) jK ( q j ) zL , q j (cid:19) (28)is the solution (Jacobi elliptic function) to Eqn.(26) forthe Dirichlet boundary conditions. One-particle densitymatrix, for the interacting Bose system, thus takes theform, from the definition, as ρ ( z, z ′ ; β ) = ∞ X j =1 (cid:20) ~ q j j K ( q j ) mL g ¯ N β [ ¯ E j − µ ] − × sn (cid:18) jK ( q j ) zL , q j (cid:19) sn (cid:18) jK ( q j ) z ′ L , q j (cid:19)(cid:21) . (29)Cluster integral of size ν = 1 , , , ... can now be easilywritten, for the interacting Bose system, by looking atthe Eqn.(21), as h ν = ∞ X j =1 e − ν ¯ E j . (30)It is easy to check, that, all these results are matchingwith the cases of ideal Bose gas in the 1-D box for g → s z = 1 / − /
2) in the same situation as above. Pair interac-tions are now possible between the particles of s z = 1 / s z = − / N in Eqns.(26) - (30) would be re-placed by ¯ N / n j = β [ ¯ Ej − µ ] − ) inEqn.(29) is to be replaced by the Fermi-Dirac statistics(¯ n j = β [ ¯ Ej − µ ] +1 ) and a phase factor ( − ν − would bemultiplied in the r.h.s. of Eqn.(30). VI. DISCUSSION AND CONCLUSION
We have analytically obtained statistical mechanicsdensity matrices for a single particle in 1-D and 3-D box.There-from we have calculated 1-particle density matri-ces for 3-D ideal Bose and Fermi gases in the box geome-tries some times by removing the boundaries along two( x − y ) axes. We also have obtained quantum cluster ex-pansions of the grand free energies in closed forms for thesame systems in the restricted geometries. There-fromwe have obtained equations of states for 3-D ideal Boseand Fermi gases in the box geometries. We also haveobtained analytic expressions for the 1-particle densitymatrices for interacting Bose and Fermi gas confined inquasi 1-D boxes. Thermodynamics of the Bose or Fermigas in the box geometry can be studied from the clusterexpansion obtained by us.Our results are novel, exact, and are directly useful fora postgraduate course on statistical mechanics or that onmany-body physics. All the calculations are done withinthe scope of undergraduate and postgraduate students.Our result would be relevant in the context of study spa-tial correlations in ultra-cold systems of dilute Bose andFermi gases of alkali atoms in 3-D magneto-optical boxtrap with quasi-uniform potential in it [23].By density matrix, we mean: statistical mechanicalunnormalized density matrix for systems in thermody-namic equilibrium. Replacing β by it/ ~ , where t is thetime taken by a particle in the system to reach ~r ′ start-ing from the position ~r at t = 0, we get the quantum mechanical density matrix-elements for all the cases.All our results can be generalized for non-ideal cases of3-D Bose and Fermi systems in box geometries at leastwithin the perturbative formalisms. Our results can alsobe generalized for d -dimensional box geometries. We arekeeping these tasks as open problems to the postgraduatestudents. Acknowledgement
Soumi Dey, Srijit Basu and Debshikha Banerjeeacknowledge partial financial support of the UGC-Networking Resource Centre, School of Physics, Univer-sity of Hyderabad, India. [1] J. von Neumann, G¨ottinger Nachrichten , 245(1927)[2] P. A. M. Dirac, Proc. Cambridge Phil. Soc. , 62 (1929); , 240 (1931)[3] R. C. Tolman, The Principles of Statistical Mechanics , p.327-361 (ch. 9), Clarendon Press, Oxford, Great Britain(1938)[4] R. P. Feynman,
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