Skip to main content
SpringerLink
Log in

Effects of quantum confinement on thermodynamic properties

  • Regular Article - Topical Issue
  • Published:
The European Physical Journal D Aims and scope Submit manuscript

Abstract

Confined quantum systems can present anomalous behaviour. In particular, thermodynamic properties such as specific heat can show special features when the system is subject to spatial confinement described, for instance, by a harmonic potential. The energy eigenvalues of this confined system can be obtained from a variational approach by using, as trial functions, the solution of a particle in a box multiplied by a gaussian. For a strong confinement regime, the energy eigenvalues converge to the same values of the particle in the box and, for a weak confinement regime, the energy eigenvalues converge to the free harmonic oscillator. This behaviour reflects in the thermodynamics properties. In the curves of specific heat as a function of temperature, for instance, it is possible to identify two regions, one when the contribution of the harmonic oscillator is dominant and the other one where the contribution of the particle in a box becomes more relevant. These results indicate a phase transition of second-order close to the Einstein temperature.

Graphic abstract

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data availability statement

This manuscript has no associated data, or the data will not be deposited. [Authors’ comment: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.].

References

  1. J.R. Sabin, E.J. Brandas, Advances in Quantum Chemistry: Theory of Confined Quantum Systems-Part One (Academic Press, New York, 2009)

    Google Scholar 

  2. J.P. Connerade, V.K. Dolmatov, P.A. Lakshmi, J. Phys. B: At. Mol. Opt. Phys. 33, 251 (2000)

    Article  ADS  Google Scholar 

  3. T.L. Cottrell, Trans. Faraday Soc. 47, 337 (1951)

    Article  Google Scholar 

  4. H. de Oliveira Batael, E. Drigo Filho, Theor Chem Acc 137, 65 (2018)

    Article  Google Scholar 

  5. H. de Oliveira Batael, E. Drigo Filho, Theor Chem Acc 139, 129 (2020)

    Article  Google Scholar 

  6. W.F. Gtari, B. Tangour, Int. J. Quantum Chem. 113, 2397 (2013)

    Article  Google Scholar 

  7. H. Lv et al., Sci. Rep. 5, 13234 (2015)

    Article  ADS  Google Scholar 

  8. N. Aquino, V. Granados, H. Yee, Rev. Mex. Fís. 55, 125 (2009)

    Google Scholar 

  9. V.C. Aguilera-Navarro, E. Ley-Koo, A.H. Zimerman, J. Phys. A: Math. Gen. 13, 3585 (1980)

    Article  ADS  Google Scholar 

  10. N. Aquino, E. Cruz, Rev. Mex. Fis. 63, 580–584 (2017)

    Google Scholar 

  11. J.S. Baijal, K.K. Singh, Progr. Theoret. Phys. 14, 214 (1955)

    Article  ADS  MathSciNet  Google Scholar 

  12. M.M. Almeida, M.N. Guimaraes, F.V. Prudente, Rev. Bras. Ensino Fis. 27, 395 (2005)

    Article  Google Scholar 

  13. G. Campoy, N. Aquino, V.D. Granados, J. Phys. A: Math. Gen 35, 4903 (2002)

    Article  ADS  Google Scholar 

  14. Lj Stevanovi, K.D. Sen, J. Phys. B: At. Mol. Opt. Phys. 41, 225002 (2008)

    Article  ADS  Google Scholar 

  15. H.E. Montgomery Jr., G. Campoy, N. Aquino, Phys. Scr. 81, 045010 (2010)

    Article  ADS  Google Scholar 

  16. L.I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968)

    Google Scholar 

  17. Doyeol Ahn, S.L. Chuang, Applied physics letters. 49, 1450 (1986)

    Article  ADS  Google Scholar 

  18. G.B. Arfken, H.J. Weber, Mathematical methods for physicists, 5th edn. (Academic Press, San Diego, 2001)

    MATH  Google Scholar 

  19. C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986)

    MATH  Google Scholar 

  20. F. Mandl, Statistical Physics, 2nd edn. (Wiley, Chilchister, 1991)

    Google Scholar 

  21. A. Riganelli, F.V. Prudente, A.J. Varandas, J. Phys. Chem. A 105, 9518–9521 (2001)

    Article  Google Scholar 

  22. M. Buchowiecki, Phys. Chem. A 124, 4048–4052 (2020)

    Article  Google Scholar 

  23. H.B. Rosenstock, Am. J. Phys. 30, 38–40 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  24. C.A. Pizarro, C.A. Condat, P.W. Lamberti, D.P. Prato, American Journal of Physics 64, 736 (1996)

    Article  ADS  Google Scholar 

  25. V. Granados, N. Aquino, Am. J. Phys. 67, 450–451 (1999)

    Article  ADS  Google Scholar 

  26. V.G. Gueorguiev, A.R.P. Rau, J.P. Draayer, American journal of physics 74, 394 (2006)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

HOB would like to thank National Council for Scientific and Technological Development (CNPq - Grant Process No. 164944/2018-4) for their financial support.

Author information

Authors and Affiliations

  1. Physics Department, São Paulo State University (UNESP), Institute of Biosciences, Humanities and Exact Sciences, São Paulo, São José do Rio Preto, 15054-000, Brazil

    Hugo de Oliveira Batael, Elso Drigo Filho & Jorge Chahine

  2. Physics Department, Federal Institute of São Paulo (IFSP), São Paulo, Votuporanga, 15503-110, Brazil

    Josimar Fernando da Silva

Authors
  1. Hugo de Oliveira Batael
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elso Drigo Filho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jorge Chahine
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Josimar Fernando da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hugo de Oliveira Batael.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Oliveira Batael, H., Drigo Filho, E., Chahine, J. et al. Effects of quantum confinement on thermodynamic properties. Eur. Phys. J. D 75, 52 (2021). https://doi.org/10.1140/epjd/s10053-021-00057-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjd/s10053-021-00057-z

Search

Navigation