Operators complex conjugation

By comparing Eq. (C.6) with Eqs. (C.2) and (C.3), the time-reversal operator can be expressed as a product of an unitary and a complex conjugate operators as follows... 

One can also define orthogonal operators, if a complex-conjugation operator has been defined on the vector space. The last three classifications are basis-set specific and cannot be meaningfully applied to general operators. 

Therefore, at this level (with spin suppressed) the time-reversal operator is just the complex conjugation operator. it which replaces i by —i. 

It was shown in Secion 13.2 that for motion in which spin is neglected, the time-reversal operator 0 is just the complex conjugation operator Y . Therefore... 

The problem treated in this paper is complicated by the fact that the linear operator T is supposed to be linearly defined on a linear space A = F of all complex functions F = F(X) of a real composite variable X = jcl5 x2, , xN, of which the L2 Hilbert space is only a small subspace. We will refer to this space A = F as the definition space of the operator T. Since it contains all complex functions, it is stable under complex conjugation, and, according to Eq. (1.50) one has T F = (TF ), which means that also the complex conjugate operator T is defined on this space. 

Therefore T (t)) obeys the time-reversed Schrodinger equation of motion. The time-reversal operator 9 is the complex conjugation operator u. 

This form of the trial function can be substituted into equation 12.24 and expanded into several terms, keeping in mind that the ordering of the basis functions is important because of the complex conjugate operation ... 

In addition there are two symmetries which can be directly related to observable quantities. The charge-conjugation symmetry, which can be expressed as H being invariant under t// (J2K1I/ (K complex conjugate operator), relates particle and hole states and thus is re-... 

In the nonrelativistic Hamiltonian we need not worry about spin. With r being real and p purely imaginary in our standard representation, the complex conjugation operator, /Co, can be used to effect time reversal. For b and d real numbers, the effect of Ko is... 

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