Calculation of Characteristic Values for Periodic Potentials

Harold D. Koenig

doi:10.1103/PhysRev.44.657

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Calculation of Characteristic Values for Periodic Potentials

Harold D. Koenig

University of Michigan

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Phys. Rev. 44, 657 – Published 15 October, 1933

DOI: https://doi.org/10.1103/PhysRev.44.657

Abstract

Section II reviews the general properties of the solutions of differential equations with periodic coefficients, the socalled Hill and Mathieu equations. In Section III asymptotic formulae for the eigenvalues of the Mathieu equation in the oscillatory region (Eq. (9)) are obtained by applying the Schrödinger perturbation theory to the problem of the physical pendulum. Section IV applies the W. K. B. method to the Hill equations. Implicit equations for the eigenvalues are obtained (§9, Eq. (28), (30) §10). They are valid for $E < V_{max}$ . In the introduction (Section I) a group of physical problems are mentioned, to which the results may be applied.

References (24)

P. Epstein, Enc. der Math. Wiss. Vol. 5, p. 507
F. Bloch, Zeits. f. Physik 52, 555 (1929) Ph. Morse, Phys. Rev. 35, 1310 (1930)
E. U. Condon, Phys. Rev. 31, 891 (1928)
L. Pauling, Phys. Rev. 31, 430 (1930)
H. Nielsen, Phys. Rev. 40, 445 (1932)
Ince, Proc. Roy. Soc. Edinburgh 46, 20, 316 (1926) ibid.47, 294 (1927) J. London Math. Soc. 2, 46 (1927) Goldstein, Trans. Camb. Phil. Soc. 23, 303 (1927)
Whittaker and Watson, Modern Analysis, Chapter 19
Ince, Differential Equations
M. J. O. Strutt, Lamé sche, Mathieusche und verwandte Funktionen in Physik und Technik, Springer, Berlin, 1932
Omitted endnote
Omitted endnote
Omitted endnote
L. Brillouin, Die Quantenstatistik, etc., Berlin, Springer, 1931
Omitted endnote
Omitted endnote
Ince, Ordinary Differential equation, p. 246 Nielsen ([5])
Strutt, [9]
Omitted endnote
Omitted endnote
H. A. Kramers, Zeits. f. Physik 39, 828 (1926) H. A. Kramers and G. P. Ittmann, 58, 217 (1929) A. Zwaan, Diss. Utrecht (1929) L. A. Young and G. E. Uhlenbeck, Phys. Rev. 36, 1154 (1930) D. M. Dennison and G. E. Uhlenbeck, 41, 313 (1932) Ta-You Wu, in press M. J. O. Strutt (Math. Ann. 101, 559 (1929)
Omitted endnote
Strutt ([16])
Omitted endnote
Omitted endnote

[c1] P. Epstein, Enc. der Math. Wiss. Vol. 5, p. 507

[c2] F. Bloch, Zeits. f. Physik 52, 555 (1929) Ph. Morse, Phys. Rev. 35, 1310 (1930)

[c3] E. U. Condon, Phys. Rev. 31, 891 (1928)

[c4] L. Pauling, Phys. Rev. 31, 430 (1930)

[c5] H. Nielsen, Phys. Rev. 40, 445 (1932)

[c6] Ince, Proc. Roy. Soc. Edinburgh 46, 20, 316 (1926) ibid.47, 294 (1927) J. London Math. Soc. 2, 46 (1927) Goldstein, Trans. Camb. Phil. Soc. 23, 303 (1927)

[c7] Whittaker and Watson, Modern Analysis, Chapter 19

[c8] Ince, Differential Equations

[c9] M. J. O. Strutt, Lamé sche, Mathieusche und verwandte Funktionen in Physik und Technik, Springer, Berlin, 1932

[c10] Omitted endnote

[c11] Omitted endnote

[c12] Omitted endnote

[c13] L. Brillouin, Die Quantenstatistik, etc., Berlin, Springer, 1931

[c14] Omitted endnote

[c15] Omitted endnote

[c16] Ince, Ordinary Differential equation, p. 246 Nielsen ([5])

[c17] Strutt, [9]

[c18] Omitted endnote

[c19] Omitted endnote

[c20] H. A. Kramers, Zeits. f. Physik 39, 828 (1926) H. A. Kramers and G. P. Ittmann, 58, 217 (1929) A. Zwaan, Diss. Utrecht (1929) L. A. Young and G. E. Uhlenbeck, Phys. Rev. 36, 1154 (1930) D. M. Dennison and G. E. Uhlenbeck, 41, 313 (1932) Ta-You Wu, in press M. J. O. Strutt (Math. Ann. 101, 559 (1929)

[c21] Omitted endnote

[c22] Strutt ([16])

[c23] Omitted endnote

[c24] Omitted endnote

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Calculation of Characteristic Values for Periodic Potentials

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