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arXiv:cond-mat/0606409

Ground State of a System of N Hard Core Quantum Particles in 1D Box

Yatendra Singh Jain

Published Jun 15, 2006• 2 citations
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Abstract

The ground state of a system of $N$ impenetrable hard core quantum particles in a 1-D box is analyzed by using a new scheme applied recently to study a similar system of two such particles {\it [Centl. Eur. J. Phys., 2(4), 709 (2004)]}. Accordingly, each particle of the system behaves like an independent entity represented by a {\it macro-orbital}, -a kind of pair waveform identical to that of a pair of particles moving with ($q$, $-q$) momenta at their {\it center of mass} which may have any momentum $K$ in the laboratory frame. It concludes: (i) $ = 0$, (ii) $ \ge λ/2$ and (iii) $q \ge q_o (= π/d)$ (with $d = L/N$ being the average nearest neighbour distance), {\it etc.} While all bosons in their ground state have $q = q_o$ and $K = 0$, fermions have $q= q_o$ with different $K$ ranging between 0 and $K = K_F$ (the Fermi wave vector). Independent of their bosonic or fermionic nature, all particles in the ground state define a close packed arrangement of their equal size wave packets representing an ordered state in phase ($ϕ-$)space with $Δϕ= 2nπ$ (with $n$ = 1,2,3, ...), $ = λ/2 = d$, and $q = q_o$. As such our approach uses greatly simplified mathematical formulation and renders a visibly clear picture of the low energy states of the systems and its results supplement earlier studies in providing their complete understanding.

DOI
10.48550/arxiv.cond-mat/0606409

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The ground state of a system of NNN impenetrable hard core quantum particles in a 1-D box is analyzed by using a new scheme applied recently to study a similar system of two such particles {\it [Centl. Eur. J. Phys., 2(4), 709 (2004)]}. Accordingly, each particle of the system behaves like an independent entity represented by a {\it macro-orbital}, -a kind of pair waveform identical to that of a pair of particles moving with (qqq, −q-q−q) momenta at their {\it center of mass} which may have any momentum KKK in the laboratory frame. It concludes: (i) =0 = 0=0, (ii) ≥λ/2 \ge λ/2≥λ/2 and (iii) q≥qo(=π/d)q \ge q_o (= π/d)q≥qo​(=π/d) (with d=L/Nd = L/Nd=L/N being the average nearest neighbour distance), {\it etc.} While all bosons in their ground state have q=qoq = q_oq=qo​ and K=0K = 0K=0, fermions have q=qoq= q_oq=qo​ with different KKK ranging between 0 and K=KFK = K_FK=KF​ (the Fermi wave vector). Independent of their bosonic or fermionic nature, all particles in the ground state define a close packed arrangement of their equal size wave packets representing an ordered state in phase (ϕ−ϕ-ϕ−)space with Δϕ=2nπΔϕ= 2nπΔϕ=2nπ (with nnn = 1,2,3, ...), =λ/2=d = λ/2 = d=λ/2=d, and q=qoq = q_oq=qo​. As such our approach uses greatly simplified mathematical formulation and renders a visibly clear picture of the low energy states of the systems and its results supplement earlier studies in providing their complete understanding.
Abstract The wave mechanics of two impenetrable hard core particles in a 1-D box is analyzed. Each particle in the box behaves like an independent entity represented by a macro-orbital (a kind of pair waveform). While the expectation value of their interaction, 〈 V HC(x) 〉, vanishes for every state of two particles, the expectation value of their relative separation, 〈 x 〉, satisfies 〈 x 〉≥λ/2 (or q ≥ π/d, with 2d=L being the size of the box). The particles in their ground state define a close-packed arrangement of their wave packets (with 〈 x 〉= λ/2, phase position separation Δϕ = 2π and momentum |q o| = π/d) and experience a mutual repulsive force (zero point repulsion) f o=h 2/2md 3 which also tries to expand the box. While the relative dynamics of two particles in their excited states represents usual collisional motion, the same in their ground state becomes collisionless. These results have great significance in determining a correct microscopic understanding of widely different many-body systems.
Macro-orbital representation of a particle (detailed account given in cond-mat/0603784) has been used to develop the microscopic theory of a system of interacting bosons. It concludes that: (i) below certain temperature (say, TλT_λTλ​), particles assume a state of (q, -q) bound pairs, (ii) the λ−λ-λ−transition is a consequence of inter-particle quantum correlations clubbed with zero-point repulsion and inter-particle attraction and represents an onset of the order-disorder of particles in their ϕ−ϕ-ϕ−space followed simultaneously by their BEC as (q, -q) bound pairs in a state of q = qo=π/dq_o = π/dqo​=π/d and K = 0, (iii) particles at T≤TλT \le T_λT≤Tλ​ acquire collective binding which locks them at = 0, = λ/2λ/2λ/2 and Δϕ=2nπΔϕ= 2nπΔϕ=2nπ (with n = 1, 2, 3, ...), (iv) the entire system assumes mechanical strain in inter-particle bonds and behaves like a single macroscopic molecule, (v) there exists an energy gap between the superfluid and normal fluid phases of the system, (vi) the λ−λ-λ−transition represents the twin phenomena of broken gauge symmetry and phase coherence, (vii) the system does not have p = 0 condensate, (viii) a new kind of quantum quasi-particle "omon" (a phononlike wave of the oscillations of the momentum coordinates of particles) exists in superfluid phase, etc. It explains the properties of He-II, including the origin of quantized vortices, critical velocities, logarithmic singularity of specific heat, etc. at quantitative level and provides microscopic foundation to two fluid theory, Ψ−Ψ-Ψ−theory, idea of macroscopic wave function, etc. The framework of the theory can unify the physics of interacting bosons and fermions.
Basic problems of a microscopic theory of many body quantum systems and different aspects of a new approach which can help in solving them are discussed in detail. To this effect we make a critical study of the wave mechanics of two hard core quantum particles and discover its several untouched aspects, viz.: (i) the useful details of ψ_k(r) (representing the relative motion of two particles), (ii) the expectation value of hard core (HC) repulsion (), (iii) the inconsistency of the statements, r \leσand ψ_k(r \leσ)=0 (σ=HC diameter of a particle), with uncertainty principle particularly for low k values, (iv) the lower bound of allowed values of k=2q, (v) the dominance of interparticle phase correlation in low temperature phase. For the first time this study concludes that has zero value which does not agree with its non-zero value known for the last several decades. This also finds compelling reasons for a system of interacting bosons such as liquid ^4He to have (q, -q) pair condensation with allowed q, obviously controlled by V_{HC}(r), to satisfy q \geπ/d. Several important aspects of N body quantum systems like liquids ^4He and ^3He are also concluded. Free from any error [see editor's note J. Scientific Exploration 16(1), p.1 (2002)], our approach can help in developing nearly exact microscopic theories of widely different systems of interacting bosons and fermions, as demonstrated for liquids ^4He type systems [J. Scientific Exploration, 16, 77-116 (2002)]. The paper also sums up the expert observations with our response to facilitate one to have a critical assessment and better understanding of the new approach.
The one-particle density matrices for hard-core bosons in a one-dimensional harmonic trap are computed numerically for systems with up to 160 bosons. Diagonalization of the density matrix shows that the many-body ground state is not Bose-Einstein condensed. The ground-state occupation, the amplitude of the lowest natural orbital, and the zero momentum peak height scale as powers of the particle number, and the corresponding exponents are related to each other. Close to its diagonal, the density matrix for hard-core bosons is similar to that of noninteracting fermions.
The London ground-state energy formula as a function of number density for a system of identical boson hard spheres, corrected for the reduced mass of a pair of particles in a "sphere-of-influence" picture, and generalized to fermion hard-sphere systems with two and four intrinsic degrees of freedom, has a double-pole at the ultimate regular (or periodic, e.g., face-centered-cubic) close-packing density usually associated with a crystalline branch. Improved fluid branches are constructed based upon exact, field-theoretic perturbation-theory low-density expansions for many-boson and many-fermion systems, extrapolated to intermediate densities via Padé and other approximants, but whose ultimate density is irregular or random closest close-packing as suggested in studies of a classical system of hard spheres. Results show substantially improved agreement with the best available Green-function Monte Carlo and diffusion Monte Carlo simulations for bosons, as well as with ladder, variational Fermi hypernetted chain, and so-called L -expansion data for two-component fermions.
We study the ground state Ψ 0 = ∑ a X | X ⟩ of N hard-core bosons on a finite lattice in configuration space, X = { x 1 ,..., x N }. All a X being positive, the ratios a X /∑ a Y can be interpreted as probabilities P a ( X ). Let E 0 denote the energy of the ground state and |∂ X | the number of nearest-neighbour particle-hole pairs in the configuration X . We prove the concentration of P a onto X with |∂ X | in a (| E 0 |) 1/2 -neighbourhood of | E 0 |, show that the average of a X over configurations with |∂ X | = n increases exponentially with n , discuss fluctuations about this average, derive upper and lower bounds on E 0 and give an argument for off-diagonal long-range order in the ground state.