Abstract.
Integral identities for particular Bloch functions in finite periodic systems are derived. All following statements are proven for a finite domain consisting of an integer number of unit cells. It is shown that matrix elements of particular Bloch functions with respect to periodic differential operators vanish identically. The real valuedness, the time-independence and a summation property of the expectation values of periodic differential operators applied to superpositions of specific Bloch functions are derived.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Bloch, ZS. f. Phys. 52, 555 (1929)
C. Kittel, Quantum Theory of Solids (John Wiley & Sons, Inc., New York, 1987)
Strictly speaking, at the band edge energies (i.e. when q∈0,π/d) the general solution [18] is either ΨB n,(q)(x)=exp (iqx)[α1 u(1) n,q(x)+α2 (xu(1) n,q(x)+u(2) n,q(x))] or ΨB n,(q)(x)=exp (iqx)[α1 u(1) n,q(x)+α2 u(2) n,q(x)], depending on the symmetry of the unit cell. Here u(1) n,q(x) and u(2) n,q(x) are two different periodic functions belonging either both to q=0 or both to q=π/d
D.W.L. Sprung, H. Wu, J. Martorell, Am. J. Phys. 61, 1118 (1993)
D.W.L. Sprung, J.D. Sigetich, H. Wu, J. Martorell, Am. J. Phys. 68, 715 (2000)
S.Y. Ren, Phys. Rev. B 64, 035322 (2001)
P. Pereyra, E. Castillo, Phys. Rev. B 65, 205120 (2002)
S.Y. Ren, Ann. Physics 301, 22 (2002)
C. Pacher, E. Gornik, Phys. Rev. B 68, 155319 (2003)
C. Pacher, W. Boxleitner, E. Gornik, Phys. Rev. B 71, 125317 (2005)
P. Pereyra, Ann. Physics 320, 1 (2005)
The so-called surface states that arise in systems with CQC lie in general energetically outside the band and are characterized by an imaginary Bloch wave number. The history of the study of surface states together with new insights have been reported recently in [11]
C. Pacher, C. Rauch, G. Strasser, E. Gornik, F. Elsholz, A. Wacker, G. Kießlich, E. Schöll, Appl. Phys. Lett. 79, 1486 (2001)
M. Reed, B. Simon, Analysis of Operators (Methods of Modern Mathematical Physics), Vol. IV (Academic Press, San Diego, 1978)
L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics: Quantum Mechanics (Non-Relativistic Theory), Vol. III 3rd edn. (Pergamon Press, Oxford, 1977)
G.L. Bir, G.E. Pikus, Symmetry and strain-induced effects in semiconductors (John Wiley & Sons, Inc., New York, 1974)
L. Jansen, M. Boon, Theory of Finite Groups (North Holland, Amsterdam, 1967)
M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic Press Ltd., Edinburgh, 1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pacher, C., Peev, M. Vanishing integral relations and expectation values for Bloch functions in finite domains. Eur. Phys. J. B 59, 519–525 (2007). https://doi.org/10.1140/epjb/e2007-00308-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2007-00308-y