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.
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References
E.H. Lieb and W. Liniger: “Exact analysis of an interacting bose gas, I. The general solution and the ground state”, Phys. Rev., Vol. 130, (1963), pp. 1605–1616.
M. Girardeau: “Relationship between systems of impenetrable bosons and fermions in one dimension”, J. Math. Phys., Vol. 1, (1960), pp. 516–523.
C.N. Yang: “Some exact results for the many-body problem in one dimension with repulsive delta function interaction”, Phys. Rev. Lett., Vol. 19, (1967), pp. 1312–1315.
Y.S. Jain: “Untouched aspects of the wave mechanics of two particles in a many body quantum system”, J. Sc. Explor., Vol. 16, (2002), pp. 67–75.
Y.S. Jain: “Basic problems of microscopic theories of many body quantum system”, cond-mat/0208445 (www.arXiv.org), pp. 1–9.
Y.S. Jain: “Microscopic theory of a system of interacting bosons: a unified approach”, J. Sc. Explor., Vol. 16, (2002),pp. 77–115.
Y.S. Jain: “Unification of the physics of interacting bosons and fermions through (q,-q) pair correlation”, J. Sc. Explor., Vol. 16, (2002),pp. 117–124.
Motions of P1 and P2 (HC size, σ) relativer to their CM can be represented by a superposition of a plane wave of momentumq with one of momentum −q (a reflected wave fromV HC (x)). Correcting such a waveform,v k (x)=sin (qx)=sin (kx/2), for σ size, we getw′ k (x CM (1)≥σ/2)=sin [k(x CM (1)−σ/2)] (withw′ k (x CM (1)<σ/2)=0) for P1 andw″ k (x CM (2)≤−σ/2)=sin [k(|x CM (2)|−σ/2)]) (withw″ k (x CM (2)>−σ/2)=0) for P2. One can express bothw′ andw″ by a single waveformw k (|x|≥σ)=sin[k(|x|−σ)/2] withw k (|x|<σ)=0 whith in the limit σ→0 becomesw k (x) = sin (k|x|/2) = |sin(kx/2)| =φ k (x)+; here we use Eqn. 8 withx=x CM (1)−x CM (2). Note thatw k (|x|<σ)=0 holds good if the occupancy of space by P1 and P2 is identified with the points occupied by the centres of their HC spheres, but the fact remains that all points (excludingx=0) covered by |x|<σ remain occupied by P1 and P2 when these centres are at |x|=σ.
P. Kleban: “Excluded volume conditions in quasi-particle theories of superfluidity”, Phys. lett., Vol. 49A, (1974), pp. 19–20.
L.I. Schiff: Quantum Mechanics, 3rd ed. McGraw Hill, New York, 1968.
K. Huang: Statistical Mechanics, Wiley Eastern Limited, New Delhi, 1991.
C.W. Woo: “Microscopic calculations for condensed phases of helium,” in The Physics of Liquid and Solid Helium, Part-I, Editors: K.H. Benneman and J.B. Ketterson, Wiley, New York, 1976, pp. 349–501.
Y.S. Jain: “Ground State of a System of N Hard Core Particles in 1-D Box”, Tech. Rep. No. PHYS./SSP-03(2003), pp. 1–21.
Y.S. Jain: “Untouched Aspects of the Wave Mechanics of a Particle in 1-D Box”, Tech. Rep. No. PHYS./SSP-01(2002), pp. 1–5.
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Jain, Y.S. Wave mechanics of two hard core quantum particles in A 1-D box. centr.eur.j.phys. 2, 709–719 (2004). https://doi.org/10.2478/BF02475571
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DOI: https://doi.org/10.2478/BF02475571