Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 5.2, p. 650   | 1 | 2 |

Section 5.2.9. Translational invariance

A. F. Moodie,a J. M. Cowleyb and P. Goodmanc

aDepartment of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia,bArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, and cSchool of Physics, University of Melbourne, Parkville, Australia

5.2.9. Translational invariance

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An important result deriving from Bethe's initial analysis, and not made explicit in the preceding formulations, is that the fundamental symmetry of a crystal, namely translational invariance, by itself imposes a specific form on wavefunctions satisfying Schrödinger's equation.

Suppose that, in a one-dimensional description, the potential in a Hamiltonian [{\bf H}_{t}(x)] is periodic, with period t. Then, [\varphi (x + t) = \varphi (x)]and [{\bf H}_{t}\psi (x) = {\bf E}\psi (x).]Now define a translation operator [\boldGamma f(x) = f(x + t),]for arbitrary [f(x)]. Then, since [\boldGamma \varphi (x) = \varphi (x)], and [\nabla^{2}] is invariant under translation, [\boldGamma {\bf H}_{t}(x) = {\bf H}_{t}(x)]and [\boldGamma {\bf H}_{t}(x)\psi (x) = {\bf H}_{t}(x + t)\psi (x + t) = {\bf H}_{t}(x) \boldGamma \psi (x).]

Thus, the translation operator and the Hamiltonian commute, and therefore have the same eigenfunctions (but not of course the same eigenvalues), i.e. [\boldGamma \psi (x) = \alpha \psi (x).]This is a simpler equation to deal with than that involving the Hamiltonian, since raising the operator to an arbitrary power simply increments the argument [\boldGamma^{m} \psi (x) = \psi (x + mt) = \alpha^{m} \psi (x).]But [\psi (x)] is bounded over the entire range of its argument, positive and negative, so that [|\alpha| = 1], and [\alpha] must be of the form [\exp\{i2\pi kt\}].

Thus, [\psi (x + t) = \boldGamma \psi (x) = \exp \{i2\pi kt\}\psi (x)], for which the solution is [\psi (x) = \exp \{i2\pi kt\} q(x)]with [q(x + t) = q(x)].

This is the result derived independently by Bethe and Bloch. Functions of this form constitute bases for the translation group, and are generally known as Bloch functions. When extended in a direct fashion into three dimensions, functions of this form ultimately embody the symmetries of the Bravais lattice; i.e. Bloch functions are the irreducible representations of the translational component of the space group.

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