An ideal quantum gas in a finite-sized container

American Journal of Physics

Volumn 66, Issue 12, 1998, Pages 1080-1085

  Pathria, R K ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032352452     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.19048     Document Type: Article

References (22)

1. G. Gutiérrez and J. M. Yáñez, "Can an ideal gas feel the shape of its container?," Am. J. Phys. 65, 739-743 (1997).
2. H. P. Baltes and E. R. Hilf, Spectra of Finite Systems (Wissenschaftsverlag, Zürich, 1976). Unfortunately, this monograph is now out of print; nevertheless, Professor Baltes informs me that he still has a few personal copies left that can be made available to interested colleagues. Professor Baltes' e-mail address is: baltes@iqe.phys.ethz.ch
3. (d-1)/2{(d-1)/2}!] + ⋯ .
4. 1/2+⋯ , A being the area of the domain and P its perimeter, appears in a recent paper by R. W. Robinett, "Visualizing classical periodic orbits from the quantum energy spectrum via the Fourier transform: Simple infinite well examples," Am. J. Phys. 65, 1167-1175 (1997).
5. See Ref. 3, Sec. 3.5.
6. The validity of replacing the sum in (5) by an integral was looked at critically by Stutz, who employed the Euler-Maclaurin formula instead. Coupled with a Fourier transformation, his calculation ended up with Eqs. (15) and (A3) of the present paper. Stutz's investigation, however, pertained to a classical gas only and his motive essentially was to establish criteria that would allow the neglect of terms other than the one with q = 0 in Eq. (15) [or the one with r = 0 in Eq. (A3)]. For details, see C. Stutz, "On the validity of converting sums to integrals in quantum statistical mechanics," Am. J. Phys. 36, 826-829 (1968).
7. See also K. Fox, "Comment on: 'On the validity of converting sums to integrals in quantum statistical mechanics,' " Am. J. Phys. 39, 116-117 (1971); A. J. Markworth, "Some comments on the conversion of sums to integrals," ibid. 39, 1400-1401 (1971).
8. See also K. Fox, "Comment on: 'On the validity of converting sums to integrals in quantum statistical mechanics,' " Am. J. Phys. 39, 116-117 (1971); A. J. Markworth, "Some comments on the conversion of sums to integrals," ibid. 39, 1400-1401 (1971).
9. See, for instance, R. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), Sec. 9.6.
10. The first rigorous demonstration of this was provided by Fowler and Jones, who carried out the summation over ∈ appearing in Eq. (7) with the help of the Poisson summation formula (15); see R. H. Fowler and H. Jones, "The properties of a perfect Einstein-Bose gas at low temperatures," Proc. Cambridge Philos. Soc. 34, 573-576 (1938).
11. Subsequently, the problem of the finite-sized Bose gas was tackled by numerous authors, some of whom employed the same technique as Fowler and Jones while others adopted different approaches. For a full survey of these works, see the review article by R. M. Ziff, G. E. Uhlenbeck, and M. Kac, "The ideal Bose-Einstein gas, revisited," Phys. Rep. 32 (4), 169-248 (1977).
12. For later developments, see R. K. Pathria, "Phase Transitions in Finite Systems," Can. J. Phys. 61, 228-238 (1983); M. N. Barber, "Finite-size scaling," in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 8, pp. 145-266, especially Sec. VI C.
13. For later developments, see R. K. Pathria, "Phase Transitions in Finite Systems," Can. J. Phys. 61, 228-238 (1983); M. N. Barber, "Finite-size scaling," in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 8, pp. 145-266, especially Sec. VI C.
14. For the two-dimensional case, which is marred by a preponderance of logarithmic terms, see R. K. Pathria, "Similarities and differences between Bose and Fermi gases," Phys. Rev. E 57, 2697-2702 (1998). The one-dimensional case, which is highly instructive, is the subject matter of the present paper.
15. L. van Hove, "Quelques propriétés générales de l'intégrale de configuration d'un systéme de particules avec interaction," Physica 15, 951-961 (1949). For a brief account of van Hove's findings, see Ref. 3, Sec. 11.1.
16. This remarkable identity was first obtained by Poisson in 1823 and is now regarded as "one of the four theta-functions transformations" of Jacobi that were established in 1828. A direct proof of this identity was given much later by Landsberg who, in 1893, derived it through contour integration; for an outline of Landsberg's proof, see E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U.P., Cambridge, 1927), Chap. VI, Example 17, p. 124.
17. See Ref. 3, Appendix D.
18. See, for instance, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), formula 1.421.4.
19. s = 2 1/(s - 1) = 3t/π, which is consistent with Eq. (28b).
20. The standard form of the Poisson summation formula pertains to the case a = - ∞, b = ∞, with f(a) and f(b) necessarily zero; see L. Schwartz, Mathematics for the Physical Sciences (Addison-Wesley, Reading, MA, 1966), Chap. V. The astute reader will note that identity (15) used earlier in Sec. III is a special case of this particular version of the Poisson formula. The more general form displayed in Eq. (A3) can be established readily by the method of Landsberg quoted in Ref. 11, and has been employed previously in S. Singh and R. K. Pathria, "Privman-Fisher hypothesis on finite systems: Verification in the case of the spherical model of ferromagnetism," Phys. Rev. B 31, 4483-4490 (1985).
21. The standard form of the Poisson summation formula pertains to the case a = - ∞, b = ∞, with f(a) and f(b) necessarily zero; see L. Schwartz, Mathematics for the Physical Sciences (Addison-Wesley, Reading, MA, 1966), Chap. V. The astute reader will note that identity (15) used earlier in Sec. III is a special case of this particular version of the Poisson formula. The more general form displayed in Eq. (A3) can be established readily by the method of Landsberg quoted in Ref. 11, and has been employed previously in S. Singh and R. K. Pathria, "Privman-Fisher hypothesis on finite systems: Verification in the case of the spherical model of ferromagnetism," Phys. Rev. B 31, 4483-4490 (1985).
22. For the integration over j appearing here, see M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC, 1964), formula 29.3.84, with s = α - 2 πir (α >0).