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arXiv:quant-ph/0603233 · Open Physics

Wave mechanics of two hard core quantum particles in A 1-D box

Yatendra Singh Jain

Published Dec 1, 2004• 4 citations
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Abstract

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.

DOI
10.2478/bf02475571

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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.
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$, (ii) $ \ge 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$, 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.
Wave mechanics of a particle in 1-D box (size =d= d=d) is critically analyzed to reveal its untouched aspects. When the particle rests in its ground state, its zero-point force (FoF_oFo​) produces non-zero strain by modifying the box size from ddd to d′=d+Δdd' = d + Δdd′=d+Δd in all practical situations where the force (FaF_aFa​) restoring ddd is not infinitely strong. Assuming that FaF_aFa​ originates from a potential ∝x2\propto x^2∝x2 (xxx being a small change in ddd), we find that: (i) the particle and strained box assume a mutually bound state (under the equilibrium between FoF_oFo​ and FaF_aFa​) with binding energy ΔE=−εo′Δd/d′Δ{E} = -ε_o'Δ{d}/d'ΔE=−εo′​Δd/d′ (with εo′=h2/8md′2ε_o' = h^2/8md'^2εo′​=h2/8md′2 being the ground state energy of the particle in the strained box), (ii) the box size oscillates around d′d'd′ when the said equilibrium is disturbed, (iii) an exchange of energy between the particle and the strained box occurs during such oscillations, and (iv) the particle, having collisional motion in its excited states, assumes collisionless motion in its ground state. These aspects have desired experimental support and proven relevance for understanding the physics of widely different systems such as quantum dots, quantum wires, trapped single particle/ion, clusters of particles, superconductors, superfluids, {\it etc.} It is emphasized that the physics of such a system in its low energy states can be truly revealed if the theory incorporates FoF_oFo​ and related aspects.
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.
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.
We consider multiple collisions of quantum wave packets in one dimension. The system under investigation consists of an impenetrable wall and two hard-core particles with very different masses. The lighter particle bounces between the heavier one and the wall. Both particles are initially represented by narrow Gaussian wave packets. A complete analytical solution of this problem is presented. The idea of the method used is to decompose the two-particle wave function into a continuous superposition of terms (channels), such that the multiple collisions within each channel do not lead to subsequent entanglement between the two particles. For each channel, the time evolution of the two-particle wave function is completely determined by the motion of the corresponding classical pointlike particles; therefore the whole quantum problem is reduced to a classical calculation. The calculation based on the above method reveals the following unexpected result: The entanglement between the two particles first increases with time due to the collisions, but then it begins to decrease, disappearing completely when the light particle becomes too slow to catch up with the heavy one.