Transformation of operators

Schwarz, K.

doi:10.1107/97809553602060000639

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 295

Section 2.2.3.2. Transformation of operators

K. Schwarz^a ^*

^a Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.3.2. Transformation of operators

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In a quantum-mechanical treatment of the electronic states in a solid we have the following different entities: points in configuration space, functions defined at these points and (quantum-mechanical) operators acting on these functions. A symmetry operation transforms the points, the functions and the operators in a clearly defined way.

Consider an eigenvalue equation of operator $[{\bb A}]$ (e.g. the Hamiltonian): $[{\bb A}\varphi=a\varphi,\eqno(2.2.3.6)]$ where $[\varphi({\bf r})]$ is a function of $[{\bf r}]$ . When g acts on $[{\bf r}]$ , the function-space operator $[\widetilde{g}]$ acts [according to (2.2.3.4)] on $[\varphi]$ yielding $[\psi]$ : $[\psi=\widetilde{g}\varphi\rightarrow\varphi=\widetilde{g}^{-1}\psi.\eqno(2.2.3.7)]$ By putting $[\varphi]$ from (2.2.3.7) into (2.2.3.6), we obtain $[{\bb A}\widetilde{g}^{-1}\psi=a\widetilde{g}^{-1}\psi. \eqno(2.2.3.8)]$ Multiplication from the left by $[\widetilde{g}]$ yields $[\widetilde{g}{\bb A}\widetilde{g}^{-1}\psi=a\widetilde{g}\widetilde{g}^{-1}\psi=a\psi.\eqno(2.2.3.9)]$ This defines the transformed operator $[\widetilde{g}{\bb A}\widetilde {g}^{-1}]$ which acts on the transformed function $[\psi]$ that is given by the original function $[\varphi]$ but at position $[g^{-1}{\bf r}]$ .