Phase factor time-reversal operator

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]... 

Note that in the present derivation we avoided providing an explicit form for the inverse of the time reversal operator. As a matter of fact, while space inversion is its own inverse, applying time reversal twice may give rise to an additional phase factor, which is -b 1 for systems with an even number of electrons, but — 1 for systems with an odd number of electrons. We shall demonstrate this point later in Sect. 7.6. Hence, = , or... 

Note that the action of time reversal on the spin functions precisely corresponds to the C operator and thus is represented hy (Cj). This result may be generalized, in the sense that time reversal can be represented as the product of complex conjugation, denoted as K, and a unitary operator acting on the components of a function space, which we shall denote by the unitary matrix U. We thus write = U/f. When this operator is applied twice, it must return the same state, except possibly for a phase factor, say exp(fx). Following Wigner, we now show that the two cases = 1 are in fact the only possibilities. Hence, the phase factor can be only either H-1 (time-even state) or -1 (time-odd state) [11, Chap. 26]. Taking time reversal twice, we have... 

In alternating current, the movement of electrons alternates direction at a set sinusoidal wave frequency. In North America this frequency is 60 hertz, which means that electrons reverse the direction of their movement in the conductor 120 times each second, effectively oscillating back and forth through the wires. Because of this, the vector direction of the magnetic held around those conductors also reverses at the same rate. This oscillation requires that the phase of the cycle be factored into design and operating principles of electric motors and other machines that use alternating current. 

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