Quantum Aspects of Materials I
Quantum mechanics is indispensable for the understanding of materials. In return, solid state physics provides beautiful illustrations for the impact of quantum dynamics on allowed energy levels in a system, for wave-particle duality, and for applications
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Quantum Aspects of Materials I
Quantum mechanics is indispensable for the understanding of materials. In return, solid state physics 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. In the present chapter we will focus on Bloch’s theorem, the duality between Bloch and Wannier states, the emergence of energy bands in crystals, and the emergence of effective mass in kp perturbation theory. We will do this for onedimensional lattices, since this captures the essential ideas. Students who would like to follow up on our introductory exposition and understand the profound impact of quantum mechanics on every physical property of materials at a deeper level should consult the monographs of Callaway [23], Ibach and Lüth [86], Kittel [95] or Madelung [109], or any of the other excellent texts on condensed matter physics— and they should include courses on condensed matter physics in their curriculum!
10.1 Bloch’s Theorem Electrons in solid materials provide a particularly beautiful realization of waveparticle duality. Bloch’s theorem covers the wave aspects of this duality. From a practical perspective, Bloch’s theorem implies that we can discuss electrons in terms of states which sample the whole lattice of ion cores in a solid material. This has important implications for the energy levels of electrons in materials, and therefore for all physical properties of materials. It is useful to recall the theory of discrete Fourier transforms as a preparation for the proof of Bloch’s theorem. We write the discrete Fourier expansion for functions f (x) with periodicity a as
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_10
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10 Quantum Aspects of Materials I
f (x) =
nx . fn exp 2π i a n=−∞ ∞
(10.1)
The orthogonality relation 1 a
a 0
nx mx exp −2π i = δmn dx exp 2π i a a
(10.2)
yields the inversion fn =
1 a
a 0
nx , dx f (x) exp −2π i a
(10.3)
and substituting this back into Eq. (10.1) yields a representation of the δ-function in a finite interval of length a, ∞ 1 x − x = δ(x − x ), exp 2π in a n=−∞ a
(10.4)
or equivalently ∞
exp(inξ ) = 2π δ(ξ ).
(10.5)
n=−∞
Equation (10.4) is the completeness relation for the Fourier monomials on an interval of length a. The Hamiltonian for electrons in a lattice with periodicity a is H =
p2 + V (x), 2m
(10.6)
where the potential operator has the periodicity of the lattice, V (x) = V (x + a) = exp
i i ap V (x) exp − ap . h¯ h¯
(10.7)
This implies i i exp ap H = H exp ap , h¯ h¯
(10.8)
and therefore the eigenspace of H with eigenvalue En can be decomposed into eigenspaces of the lattice translation operator
10.1 Bloch’s Theorem
205
i T (a) = exp ap . h¯
(10.9)
The eigenvalues of this unitary operator
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