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

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) $<Aδ{(x)}> = 0$, (ii) $<x> \ge λ/2$ 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, ...), $<x> = λ/2 = d$, 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.