Time-reversal operators

Janssen, T.

doi:10.1107/97809553602060000629

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. 1.2, pp. 54-55

Section 1.2.5.4. Time-reversal operators

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.2.5.4. Time-reversal operators

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In quantum mechanics, symmetry transformations act on state vectors as unitary or anti-unitary operators. For the Schrödinger equation for one particle without spin, $[\hbar i {{\partial}\over{\partial t}}\Psi ({\bf r},t) = H\Psi ({\bf r},t), \eqno (1.2.5.2)]$ the operator that reverses time is the complex conjugation operator $[\theta]$ with $[\theta \Psi ({\bf r},t) = \Psi^{*} ({\bf r},t) \eqno (1.2.5.3)]$ satisfying $[\hbar i {{\partial}\over{\partial t}}\Psi^{*} ({\bf r},-t) = H\Psi^{*} ({\bf r},-t),]$ which is the time-reversed equation.

This operator is anti-linear [ $[\theta (\alpha \Psi +\beta \Phi)= \alpha^{*}\theta \Psi +\beta^{*}\Phi ]$ ] and has the following commutation relations with the operators $[{\bf r}]$ and $[{\bf p}]$ for position and momentum: $[\theta {\bf r} \theta^{-1} = {\bf r},\quad \theta {\bf p}\theta^{-1}. \eqno (1.2.5.4)]$ For a Euclidean transformation $[g=\{R|{\bf a}\}]$ , the operation on the state vector is given by the unitary operator $[T_{g}\Psi ({\bf r}) = \Psi (g^{-1}{\bf r}) = \Psi (R^{-1}({\bf r}-{\bf a})). \eqno (1.2.5.5)]$ The two operators $[\theta]$ and $[T_{g}]$ commute. Therefore, if g is an orthochronous element of the symmetry group, the corresponding operator is $[T_{g}]$ , and if [gT] is an antichronous element the operator is $[\theta T_{g}]$ . The operator $[\theta T_{g}]$ is also anti-unitary: it is anti-linear and conserves the absolute value of the Hermitian scalar product: $[| \langle\theta T_{g}\Psi |\theta T_{g}\Phi\rangle| = | \langle \Psi |\Phi\rangle|]$ .

If the particle has a spin, the time-reversal operator has to have the commutation relation $[\theta {\bf S} \theta^{-1} = -{\bf S} \eqno (1.2.5.6)]$ with the spin operator $[{\bf S}]$ . For a spin-½ particle, the spin operators are $[S_{i}=\hbar \sigma_{i}/2]$ in terms of the Pauli matrices. Then the time-reversal operator is $[T_{T} = \sigma_{2}\theta. \eqno (1.2.5.7)]$ The operators corresponding to the elements of a magnetic symmetry group are generally (anti-)unitary operators on the state vectors. These operators form a representation of the magnetic symmetry group. $[T_{g}T_{g'} = T_{gg'}. \eqno (1.2.5.8)]$ In principle, they even form a projective representation, but as discussed before for particles without spin the factor system is trivial, and for particles with spin one can take as the symmetry group the double group of the symmetry group.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 54-55