Translational invariance

Moodie, A. F.; Cowley, J. M.; Goodman, P.

doi:10.1107/97809553602060000570

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 5.2, p. 554 | 1 | 2 |

Section 5.2.9. Translational invariance

A. F. Moodie,^a J. M. Cowley^b ^‡ and P. Goodman^c ^‡

^a Department 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 3052

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.

References

International Tables for Crystallography (2006). Vol. B. ch. 5.2, p. 554