Abstract
Quantum mechanics is indispensable for the understanding of materials. Solid state physics, in return, provides beautiful illustrations for the impact of quantum dynamics on allowed energy levels in a system, for wave-particle duality, and for applications of perturbation theory.
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Notes
- 1.
This is a consequence of Schur’s Lemma in group theory: Abelian symmetry groups have one-dimensional irreducible representations.
- 2.
F. Bloch, Z. Phys. 52, 555 (1929). As a mathematical theorem in the theory of differential equations it is known as Floquet’s theorem due to G. Floquet, Ann. sci. de l’É.N.S., 2e série, 12, 47 (1883).
- 3.
This is exactly as in (10.1), only with periodicity 2π ∕ a.
- 4.
It is tempting to conclude that the Wannier functions w n, ν(x) should be centered around the lattice site x = νa, but this is not what generically happens. The Wannier function w n, ν(x) is usually large in a unit cell containing the lattice site x = νa, but localization around the lattice site requires inclusion of extra phase factors exp[iφ(n, k)] in the Bloch functions, see W. Kohn, Phys. Rev. 115, 809 (1959) and F.B. Pedersen, G.T. Einevoll, P.C. Hemmer, Phys. Rev. B 44, 5470 (1991).
- 5.
However, the function | w n, ν(x, t) | for higher n can have its maxima far from the lattice site νa.
- 6.
R. de L. Kronig & W.G. Penney, Proc. Roy. Soc. London A 130, 449 (1931).
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Dick, R. (2012). Quantum Aspects of Materials I. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8077-9_10
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