Antiunitary operators

Antilinear operator, antiunitary, 688 Antiunitary operators, 727 A-operation, 524 upon Dirac equation, 524 Approximation, 87 methods, successive minimax (Chebyshev), 96 problem of, 52 Arc, 258... 

Operators that induce transformations in space satisfy eq. (2) and are therefore unitary operators with the property / T = 1. An operator that satisfies eq. (3) is said to be antiunitary. In contrast to spatial symmetry operators, the time-reversal operator is anti-unitary. Let U denote a unitary operator and let T denote an antiunitary operator. 

Hence, unitary operators are linear operators, but an antiunitary operator is antilinear. 

The product of two antiunitary operators is a unitary operator. Consequently,... 

Consider the set of operators If OR, where H R is a group of unitary symmetry operators and OR is therefore a set of antiunitary operators. Since rotations and time reversal commute, the multiplication mles within this set are... 

Consider the group G = H, AH that contains unitary H and antiunitary AH operators. H is necessarily an invariant subgroup of G of index 2 and AH is a coset of H with coset representative A (which may be any one of the antiunitary operators of G) so that... 

Given the IRs T of H, all the irreducible co-representations F of G can be determined from eqs. (40)-(42). Although the equivalence of T, T and the sign of c(Z) provide a criterion for the classification of the co-representations of point groups with antiunitary operators, this will be more useftd in the form of a character test. 

Exercise 18.5-1 (a) Find the operator Jf(l 1. (b) Prove that Jf0 is an antiunitary operator, (c) Show that... 

There are two instances in which the antiunitary operator is just.// rather than A Q,... 

A antiunitary operator (Section 13.1) A, B may also denote linear,... 

In addition, Hso is time-reversal invariant. The time-reversal operator for a single electron is the antiunitary operator (up to an arbitrary phase factor) [25]... 

Consequently the spinors < that make up Q must transform according to the irreducible representations or corepresentations (if the invariance group contains antiunitary operators) of the invariance group. 

Comparing with the commutation relations above, we see that for r and p at least, K has the effect of an antiunitary operator. Expressing orbital angular momentum as f = r X p, we see that = —1. For spin we can draw on the analogies between the transformation of commutation relations for spin and orbital angular momentum. From these we see that the transformed commutation relations are consistent with Cs0 = -s, and the spin has the same transformation properties as orbital angular momentum. 

We can explore the fermion case further. Because IC is an antiunitary operator, we must have... 

Determine the effect of applying Ux to an ON vector k>. This operator will be used in Section 11.8.6, in our discussion of direct ctailiguration-interaction methods. The time-reversal operator, which changes the direction of time in quantum mechanics, is an antiunitary operator T that can be written as ... 

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