Appendix B Antilinear Operators and Their Properties Appendix C Proof of Eqs. (18) and (23) [Pg.552]

Furthermore, any antilinear operator can be written as the product of a unitary operator and the operator K. Specifically, we can write the time reversal operator as 6 = UK, and our problem is now that of [Pg.729]

This should apply both to linear and antilinear operators, hereafter denoted, respectively, by t and A. For a linear operator, the action on a general state ci /i) + C2 v /2) is expressed by [Pg.614]

That is, unitary operators are linear and anti-unitary operators are antilinear. [Pg.728]

If we assume that the adjoint A of an antilinear operator is defined as in the case of a linear operator by the equation [Pg.688]

This implies that the Hermitian conjugate of an antilinear operator is also antilinear. It should also be pointed out that the product of two antilinear [Pg.614]

In this appendix, we review some important properties of antilinear operators that are used in the text and Appendix C. Let us then consider an operator O that [Pg.613]

From the definitions (a)-(b) it follows that a product of an even number of antilinear operators is a linear operator, whereas the product of an odd number of antilinear operators is an antilinear operator. Similarly a product of any number of linear operators and an even (odd) number of antilinear operators is a linear (antilinear) operator. [Pg.688]

If A induces a one-to-one mapping of the Hilbert space on itself then the inverse operator A 1 exists. It is an antilinear operator with the property that [Pg.688]

Geometric phase effect (GPE) (Continued) adiabatic states, conical intersections invariant operators, 735-737 Jahn-Teller theorem, 733-735 antilinear operator properties, 721-723 degenerate/near-degenerate vibration levels, 728-733 [Pg.79]

Antiferromagnetic one-dimensional Kronig-Penney potentials, 747 Antiferromagnetic single particle potential, 747 energy band for, 747 Antilinear operator, 687 adjoint of, 688 antihermitian, 688 [Pg.769]

Thus Ql —0 does satisfy the correct Heisenberg equation of motion. It should be recalled, incidentally, that the correct definition of the adjoint A of an antilinear operator A is [Pg.688]

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