Skip to main content
SpringerLink
Log in

The Quantum-Mechanical Many-Body Problem: The Bose Gas

  • Chapter
Condensed Matter Physics and Exactly Soluble Models

Abstract

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4–5 decades ago, and to explore new regimes not treated before. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous—and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πћ 2aρ/m where a is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5 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 been obtained by us 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. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, several other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the ‘traps’ actually used in the experiments. For this system it is also possible to prove complete Bose condensation and superfluidity, as we have shown. On the frontier of experimental developments is the possibility that a dilute gas in an elongated trap will behave like a one-dimensional system; we have proved this mathematically. Another topic is a proof that Foldy’s 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy; using this we can also prove the N7/5 formula for the ground state energy of the two-component charged Bose gas proposed by Dyson in 1967. All of this is quite recent work and it is hoped that the mathematical methodology might be useful, ultimately, to solve more complex problems connected with these interesting systems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G.E. Astrakharchik and S. Giorgini, Quantum Monte Carlo study of the three-to one-dimensional crossover for a trapped Bose gas, Phys. Rev. A 66, 0536141–6 (2002).

    Google Scholar 

  2. G.E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Superfluidity versus Bose-Einstein condensation in a Bose gas with disorder, Phys. Rev. A 66, 023603 (2002).

    Google Scholar 

  3. B. Baumgartner, The Existence of Many-particle Bound States Despite a Pair Interaction with Positive Scattering Length, J. Phys. A 30 (1997), L741 — L747.

    Article  ADS  Google Scholar 

  4. Bm]G.Baym, in: Math. Methods in Solid State and Superfluid Theory, Scottish Univ Summer School of Physics, Oliver and Boyd, Edinburgh (1969).

    Google Scholar 

  5. ). F.A. Berezin, General concept of quantization, Commun. Math. Phys. 40, 153–174 (1975).

    Google Scholar 

  6. D. Blume, Fermionization of a bosonic gas under highly elongated confinement: A diffusion quantum Monte Carlo study, Phys. Rev. A 66, 053613–1–8 (2002).

    Google Scholar 

  7. N.N. Bogolubov, J. Phys. (U.S.S.R.) 11, 23 (1947); N.N. Bogolubov, D.N. Zubarev, Soy. Phys.-JETP 1, 83 (1955).

    Google Scholar 

  8. K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Artl, W. Ertmer, and K. Sengstok, Waveguides for Bose-Einstein condensates, Phys. Rev. A, 63, 031602 (2001)

    Google Scholar 

  9. S.N. Bose, Plancks Gesetz and Lichtquantenhypothese, Z. Phys. 26, 178–181 (1924).

    Article  MATH  Google Scholar 

  10. A.Y. Cherny, A.A. Shanenko, Dilute Bose gas in two dimensions: Density expansions and the Gross-Pitaevskii equation, Phys. Rev. E 64, 027105 (2001)

    Google Scholar 

  11. A.Y. Cherny, A.A. Shanenko, The kinetic and interaction energies of a trapped Bose gas: Beyond the mean field, Phys. Lett. A 293, 287 (2002).

    Article  ADS  MATH  Google Scholar 

  12. J. Conlon, E.H. Lieb, H.-T. Yau, The N7í5 Law for Charged Bosons, Commun. Math. Phys. 116, 417–448 (1988).

    MathSciNet  ADS  Google Scholar 

  13. S.L. Cornish, N.R. Claussen, J.L. Roberts, E.A. Cornell, C.E. Wieman, Stable 85 Ró Bose-Einstein Condensates with Widely Tunable Interactions, Phys. Rev. Lett. 85, 1795–98 (2000).

    Article  ADS  Google Scholar 

  14. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. S.ringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463–512 (1999).

    Article  ADS  Google Scholar 

  15. K.K. Das, M.D. Girardeau, and E.M. Wright, Crossover from One to Three Dimensions for a Gas of Hard–Core Bosons, Phys. Rev. Lett. 89, 110402–1–4 (2002).

    Google Scholar 

  16. V. Dunjko, V. Lorent, and M. Olshanii, Bosons in Cigar-Shaped Traps: Thomas-Fermi Regime, Tonks-Girardeau Regime, and In Between, Phys. Rev. Lett. 86, 5413–5316 (2001).

    Article  ADS  Google Scholar 

  17. F.J. Dyson, Ground-State Energy of a Hard-Sphere Gas, Phys. Rev. 106, 20–26 (1957).

    Article  MATH  Google Scholar 

  18. F.J. Dyson, Ground State Energy of a Finite System of Charged Particles, J. Math. Phys. 8, 1538–1545 (1967).

    Article  MathSciNet  ADS  Google Scholar 

  19. F.J. Dyson, E.H. Lieb, B. Simon, Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions, J. Stat. Phys. 18, 335–383 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  20. A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzber. Kgl. Preuss. Akad. Wiss., 261–267 (1924), and 3–14 (1925).

    Google Scholar 

  21. A.L. Fetter and A.A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate, J. Phys.: Condens. Matter 13, R135 (2001).

    Article  ADS  Google Scholar 

  22. D.S. Fisher, P.C. Hohenberg, Dilute Bose gas in two dimensions, Phys. Rev. B 37, 4936–4943 (1988).

    Google Scholar 

  23. L.L. Foldy, Charged Boson Gas, Phys. Rev. 124, 649–651 (1961); Errata ibid 125, 2208 (1962).

    Google Scholar 

  24. M.D. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension, J. Math. Phys. 1, 516 (1960).

    Article  ADS  Google Scholar 

  25. M.D. Girardeau and E.M. Wright, Bose–Fermi variational Theory for the BEC–Tonks Crossover, Phys. Rev. Lett. 87, 210402–1–4 (2001).

    Google Scholar 

  26. M.D. Girardeau, E.M. Wright, and J.M. Triscari, Ground–state properties of a one–dimensional system of hard–core bosons in a harmonic trap, Phys. Rev. A 63, 033601–1–6 (2001).

    Google Scholar 

  27. Go] A. Görlitz, et al., Realization of Bose–Einstein Condensates in Lower Dimen–sion,Phys. Rev. Lett. 87 130402–1–4 (2001).

    Google Scholar 

  28. G.M. Graf and J.P. Solovej, A correlation estimate with applications to quantum systems with Coulomb interactions, Rev. Math. Phys. 6, 977–997 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  29. G] M. Greiner, et al., Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,Phys. Rev. Lett. 87 160405 (2001).

    Google Scholar 

  30. E.P. Gross, Structure of a Quantized Vortex in Boson Systems, Nuovo Cimento 20, 454–466 (1961).

    Article  MATH  Google Scholar 

  31. E.P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys. 4, 195–207 (1963).

    Article  ADS  Google Scholar 

  32. D.F. Hines, N.E. Frankel, D.J. Mitchell, Hard disc Bose gas, Phys. Lett. 68A, 12–14 (1978).

    Article  Google Scholar 

  33. P.C. Hohenberg, Existence of Long-range Order in One and Two Dimensions, Phys. Rev. 158, 383–386 (1966).

    Google Scholar 

  34. HoM] P.C. Hohenberg and P.C. Martin, Microscopic theory of helium, Ann. Phys. (NY) 34, 291 (1965).

    Article  ADS  Google Scholar 

  35. K. Huang, in: Bose-Einstein Condensation, A. Griffin, D.W. Stroke, S. Stringari, eds., Cambridge University Press, 31–50 (1995).

    Google Scholar 

  36. K. Huang, C.N. Yang, Phys. Rev. 105, 767–775 (1957); T.D. Lee, K. Huang, 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, Soy. 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).

    Google Scholar 

  37. A.D. Jackson and G.M. Kavoulakis, Lieb Mode in a Quasi-One-Dimendional Bose-Einstein Condensate of Atoms, Phys. Rev. Lett. 89, 070403 (2002).

    Google Scholar 

  38. T. Kennedy, E.H. Lieb, S. S.astry, The XY Model has Long-Range Order for all Spins and all Dimensions Greater than One, Phys. Rev. Lett. 61, 25822584 (1988).

    Google Scholar 

  39. 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).

    Google Scholar 

  40. E.B. Kolomeisky, T.J. Newman, J.P. Straley, X. Qi, Low-dimensional Bose liquids: beyond the Gross-Pitaevskii approximation, Phys. Rev. Lett. 85, 11461149 (2000).

    Google Scholar 

  41. M. Kobayashi and M. Tsubota, Bose-Einstein condensation and superfluidity of a dilute Bose gas in a random potential, Phys. Rev. B 66, 174516 (2002).

    Google Scholar 

  42. S. Komineas and N. Papanicolaou, Vortex Rings and Lieb Modes in a Cylin- drical Bose-Einstein Condensate, Phys. Rev. Lett. 89, 070402 (2002).

    Google Scholar 

  43. A. Lenard, Momentum distribution in the ground stat of the one-dimensional system of impenetrable bosons, J. Math. Phys. 5, 930–943 (1964).

    Article  ADS  Google Scholar 

  44. L1] E.H. Lieb, Simplified Approach to the Ground State Energy of an Imperfect Bose Gas,Phys. Rev. 130 2518–2528 (1963). See also Phys. Rev. 133 (1964), A899–A906 (with A.Y. Sakakura) and Phys. Rev. 134 (1964), A312–A315 (with W. Liniger).

    Google Scholar 

  45. 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).

    Google Scholar 

  46. E.H. Lieb, The classical limit of quantum spin systems, Commun. Math. Phys. 31, 327–340 (1973).

    ADS  Google Scholar 

  47. E.H. Lieb, The Bose Gas: A Subtle Many-Body Problem, in Proceedings of the XIII International Congress on Mathematical Physics, London, A. Fokas, et al. eds. International Press, pp. 91–111, 2001.

    Google Scholar 

  48. E.H. Lieb, W. Liniger, Exact Analysis of an Interacting Bose Gas.I . The Gen-eral 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).

    Article  MathSciNet  MATH  Google Scholar 

  49. E.H. Lieb, M. Loss, Analysis, 2nd ed., Amer. Math. Society, Providence, R.I. (2001).

    MATH  Google Scholar 

  50. Errata J. Stat. Phys. 14, 465 (1976).

    Google Scholar 

  51. E.H. Lieb, R. Seiringer, Proof of Bose–Einstein Condensation for Dilute Trapped Gases, Phys. Rev. Lett. 88, 170409–1–4 (2002).

    Google Scholar 

  52. E.H. Lieb, R. Seiringer, J.P. Solovej, and J. Yngvason, The ground state of the Bose gas, in: Current Developments in Mathematics, 2001, 131–178, International Press, Cambridge (2002).

    Google Scholar 

  53. E.H. Lieb, R. Seiringer, J. Yngvason, Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional, Phys. Rev A 61, 043602 (2000).

    Google Scholar 

  54. LSeY2] EH. Lieb, R. Seiringer, J. Yngvason, A Rigorous Derivation of the GrossPitaevskii Energy Functional for a Two-dimensional Bose Gas, Commun. Math. Phys. 224, 17 (2001).

    ADS  Google Scholar 

  55. 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).

    Google Scholar 

  56. E.H. Lieb, R. Seiringer, J. Yngvason, Two-Dimensional Gross-Pitaevskii Theory, in: Progress in Nonlinear Science, Proceedings of the International Conference Dedicated to the 100th Anniversary of A.A. Andronov, Volume II, A.G. Litvak, ed., 582–590, Nizhny Novgorod, Institute of Applied Physics, University of Nizhny Novgorod (2002).

    Google Scholar 

  57. E.H. Lieb, R. Seiringer, J. Yngvason, Superfluidity in Dilute Trapped Bose Gases, Phys. Rev. B 66, 134529 (2002).

    Google Scholar 

  58. E.H. Lieb, R. Seiringer, J. Yngvason, One–Dimensional Behavior of Dilute, Trapped Bose Gases, Commun. Math. Phys. 244, 347–393 (2004). See also: One–Dimensional Bosons in Three–Dimensional Traps, Phys. Rev. Lett. 91, 150401–1–4 (2003).

    Google Scholar 

  59. E.H. Lieb, R. Seiringer, and J. Yngvason, Poincaré Inequalities in Punctured Domains, Ann. Math. 158, 1067–1080 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  60. E.H. Lieb, J.P. Solovej, Ground State Energy of the One-Component Charged Bose Gas, Commun. Math. Phys. 217, 127–163 (2001). Errata 225, 219–221 (2002).

    MathSciNet  Google Scholar 

  61. LSo2] E.H. Lieb, J.P. Solovej, Ground State Energy of the Two-Component Charged Bose Gas,Commun. Math. Phys. (in press), arxiv:math-ph/0311010, mp..arc 03–490.

    Google Scholar 

  62. E.H. Lieb, J. Yngvason, Ground State Energy of the low density Bose Gas, Phys. Rev. Lett. 80, 2504–2507 (1998).

    ADS  Google Scholar 

  63. E.H. Lieb, J. Yngvason, The Ground State Energy of a Dilute Two-dimensional Bose Gas, J. Stat. Phys. 103, 509 (2001).

    MathSciNet  MATH  Google Scholar 

  64. 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).

    Google Scholar 

  65. H. Moritz, T. Stöferle, M. Köhl and T. Esslinger, Exciting Collective Oscillations in a Trapped 1D Gas, Phys. Rev. Lett. 91, 250402 (2003).

    Google Scholar 

  66. M] W.J. Mullin, Bose-Einstein Condensation in a Harmonic Potential,J. Low

    Google Scholar 

  67. Temp. Phys. 106, 615–642 (1997).

    Google Scholar 

  68. M. Olshanii, Atomic Scattering in the Presence of an External Confinement

    Google Scholar 

  69. and a Gas of Impenetrable Bosons,Phys. Rev. Lett. 81 938–941 (1998).

    Google Scholar 

  70. A.A. Ovchinnikov, On the description of a two-dimensional Bose gas at low densities, J. Phys. Condens. Matter 5, 8665–8676 (1993). See also JETP Letters 57, 477 (1993); Mod. Phys. Lett. 7, 1029 (1993).

    Google Scholar 

  71. C. Pethick, H. Smith, Bose Einstein Condensation of Dilute Gases, Cambridge University Press, 2001.

    Google Scholar 

  72. D.S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven, Regimes of Quantum Degeneracy in Trapped ID Gases, Phys. Rev. Lett. 85, 3745–3749 (2000).

    Article  ADS  Google Scholar 

  73. L.P. Pitaenskii,Vortes in an imperfect Bose gas, Sov. Phys.JETP. 13, 451–454 (1961).

    Google Scholar 

  74. PiSt] L. Pitaevskii, S. Stringari, Uncertainty Principle, Quantum Fluctuations, and Broken Symmetries, J. Low Temp. Phys. 85, 377 (1991).

    Article  ADS  Google Scholar 

  75. V.N. Popov, On the theory of the superfluidity of two-and one-dimensional Bose systems, Theor. and Math. Phys. 11, 565–573 (1977).

    Google Scholar 

  76. PrSv] N.V. Prokof’ev and B.V. Svistunov, Two definitions of superfluid density, Phys. Rev. B 61, 11282 (2000).

    Article  Google Scholar 

  77. M. Schick, Two-Dimensional System of Hard Core Bosons, Phys. Rev. A 3, 1067–1073 (1971).

    Article  Google Scholar 

  78. Sc] F. Schreck, et al., Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,Phys. Rev. Lett. 87 080403 (2001).

    Google Scholar 

  79. R. Seiringer, Diplom thesis, University of Vienna, 1999.

    Google Scholar 

  80. R. Seiringer, Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground State Energy Formula, in: Partial Differential Equations and Spectral Theory, PDE2000 Conference in Clausthal, Germany, M. Demuth and B.-W. Schulze, eds., 307–314, Birkhäuser (2001).

    Google Scholar 

  81. R. Seiringer, Gross-Pitaevskii Theory of the Rotating Bose Gas, Commun. Math. Phys. 229, 491–509 (2002); Ground state asymptotics of a dilute, rotating gas, J. Phys. A: Math. Gen. 36, 9755–9778 (2003).

    Google Scholar 

  82. S.I. Shevchenko, On the theory of a Bose gas in a nonuniform field, Sov. J. Low Temp. Phys. 18, 223–230 (1992).

    MathSciNet  Google Scholar 

  83. B. Simon, 7’race ideals and their application, Cambridge University Press (1979).

    Google Scholar 

  84. So] J.P. Solovej, Upper Bounds to the Ground State Energies of the One-and Two-Component Charged Bose gases,preprint, arxiv:math-ph/0406014.

    Google Scholar 

  85. G. Temple, The theory of Rayleigh’s Principle as Applied to Continuous Sys- tems, Proc. Roy. Soc. London A 119, 276–293 (1928).

    Article  ADS  MATH  Google Scholar 

  86. D.R. Tilley and J. Tilley, Superfluidity and Superconductivity, third edition, Adam Bulger, Bristol and New York (1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, Jadwin Hall, P.O. Box 708, Princeton, NJ, 08544, USA

    Elliott H. Lieb

  2. Department of Physics, Princeton University, Jadwin Hall, P.O. Box 708, Princeton, NJ, 08544, USA

    Robert Seiringer

  3. School of Mathematics, Institute for Advanced Study, 1 Einstein Dr, Princeton, N.J., 08540, USA

    Jan Philip Solovej

  4. On Leave From Dept. of Math, University of Copenhagen, Universitetsparken 5, dk-2100, Copenhagen, Denmark

    Jan Philip Solovej

  5. Institut fur Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090, Vienna, Austria

    Jakob Yngvason

Authors
  1. Elliott H. Lieb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Seiringer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Philip Solovej
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jakob Yngvason
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of California, One Shields Ave., 95616-8633, Davis, CA, USA

    Bruno Nachtergaele

  2. Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Jan Philip Solovej

  3. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090, Wien, Austria

    Jakob Yngvason

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J. (2004). The Quantum-Mechanical Many-Body Problem: The Bose Gas. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Condensed Matter Physics and Exactly Soluble Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06390-3_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-06390-3_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-06093-9

  • Online ISBN: 978-3-662-06390-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation