Abstract
Now that the properties of the ground state of quantum-mechanical many-body systems (bosons) at low density, p, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4–5 decades ago. One of these is that the leading term in the energy/particle is 4πaρ where a is the scattering length of the 2-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N 7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained jointly with J. Yngvason and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. Another problem of great interest is the existence of Bose-Einstein condensation, and what little is known about this rigorously will also be discussed. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof (with Yngvason and Seiringer) that the Gross-Pitaevskii equation correctly describes the ground state in the ‘traps’ actually used in the experiments. The other is a very recent proof (with Solovej) that Foldy’s 1961 theory of a high density gas of charge particles correctly describes its ground state energy.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E.H. Lieb, J. Yngvason, Ground State Energy of the low density Bose Gas, Phys. Rev. Lett. 80, 2504–2507 (1998). arXiv math-ph/9712138, mp-arc 97–631
E.H. Lieb, J. Yngvason, The Ground State Energy of a Dilute Two-dimensional Bose Gas, J. Stat. Phys. (in press) arXiv math-ph/0002014, mp_arc 00–63.
E.H. Lieb, R. Seiringer, and J. Yngvason, Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional, Phys. Rev A 61, 043602–1 – 043602–13 (2000)mp_arc 99–312, arXiv math-ph/9908027 (1999).
E.H. Lieb, R. Seiringer and J. Yngvason, A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-dimensional Bose Gas, Commun. Math. Phys. (in press). arXiv cond-mat/0005026, mp_arc 00–203.
E.H. Lieb, J.P. Solovej, Ground State Energy of the One-Component Charged Bose Gas, Commun. Math. Phys. bf 217, 127–163 (2001). arXiv cond-mat/0007425, mp_arc 00–303.
E.H. Lieb, J. Yngvason, The Ground State Energy of a Dilute Bose Gas, in Differential Equations and Mathematical Physics, University of Alabama, Birmingham, 1999, R. Weikard and G. Weinstein, eds., 271–282 Amer. Math. Soc./Internat. Press (2000). arXiv math-ph/9910033, mp-arc 99–401.
E.H. Lieb, R. Seiringer, J. Yngvason, The Ground State Energy and Density of Interacting Bosons in a Trap, in Quantum Theory and Symmetries, Goslar, 1999, H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke, eds., pp. 101–110, World Scientific (2000). arXiv math-ph/9911026, mp-arc 99–439.
W. Ketterle, N. J. van Druten, Evaporative Cooling of Trapped Atoms, in B. Bederson, H. Walther, eds., Advances in Atomic, Molecular and Optical Physics, 37, 181–236, Academic Press (1996).
F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463–512 (1999).
R. Seiringer, Diplom thesis, University of Vienna, (1999).
N.N. Bogolubov, J. Phys. (U.S.S.R.) 11, 23 (1947);
N.N. Bogolubov and D.N. Zubarev, Sov. Phys.-JETP 1, 83 (1955).
K. Huang, C.N. Yang, Phys. Rev. 105, 767–775 (1957).
T.D. Lee, K. Huang, and C.N. Yang, Phys. Rev. 106, 1135–1145 (1957).
K.A. Brueckner, K. Sawada, Phys. Rev. 106, 1117–1127, 1128–1135 (1957).
S.T. Beliaev, Sov. Phys.-JETP 7, 299–307 (1958).
T.T. Wu, Phys. Rev. 115, 1390 (1959).
N. Hugenholtz, D. Pines, Phys. Rev. 116, 489 (1959).
M. Girardeau, R. Arnowitt, Phys. Rev. 113, 755 (1959).
T.D. Lee, C.N. Yang, Phys. Rev. 117, 12 (1960).
E.H. Lieb, Simplified Approach to the Ground State Energy of an Imperfect Bose Gas, Phys. Rev. 130 (1963), 2518–2528. See also Phys. Rev. 133 (1964), A899–A906 (with A.Y. Sakakura) and Phys. Rev. 134 (1964), A312–A315 (with W. Liniger).
E.H. Lieb, The Bose fluid, in W.E. Brittin, ed., Lecture Notes in Theoretical Physics VIIC, Univ. of Colorado Press, pp. 175–224 (1964).
F.J. Dyson, Ground-State Energy of a Hard-Sphere Gas, Phys. Rev. 106, 20–24 (1957).
B. Baumgartner, The Existence of Many-particle Bound States Despite a Pair Interaction with Positive Scattering Length, J. Phys. A 30 (1997), L741–L747.
E.H. Lieb, W. Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State, Phys. Rev. 130, 1605–1616 (1963);
E.H. Lieb, Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Phys. Rev. 130, 1616–1624 (1963).
G. Temple, The theory of Rayleigh’s Principle as Applied to Continuous Systems, Proc. Roy. Soc. London A 119 (1928), 276–293.
M. Schick, Two-Dimensional System of Hard Core Bosons, Phys. Rev. A 3, 1067–1073 (1971).
T. Kennedy, E.H. Lieb, S. Shastry, The XY Model has Long-Range Order for all Spins and all Dimensions Greater than One, Phys. Rev. Lett. textbf 61, 2582–2584 (1988).
E.H. Lieb, H. Narnhofer, The Thermodynamic Limit for Jellium, J. Stat. Phys. 12, 291–310 (1975). Errata J. Stat. Phys. 14, 465 (1976).
L.L. Foldy, Charged Boson Gas, Phys. Rev. 124, 649–651 (1961); Errata ibid 125, 2208 (1962).
F.J. Dyson, Ground State Energy of a Finite System of Charged Particles, J. Math. Phys. 8, 1538–1545 (1967).
J. Conlon, E.H. Lieb, H-T. Yau The N 7/5 Law for Charged Bosons, Commun. Math. Phys. 116, 417–448 (1988).
G.M. Graf, J.P. Solovej, A correlation estimate with applications to quantum systems with Coulomb interactions, Rev. Math. Phys., 6 (No. 5a, Special Issue) 977–997 (1994). mp_arc 93–60.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lieb, E.H. (2001). The Bose Gas: A Subtle Many-Body Problem. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_46
Download citation
DOI: https://doi.org/10.1007/978-3-662-04360-8_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-04362-2
Online ISBN: 978-3-662-04360-8
eBook Packages: Springer Book Archive