Time reversal with spin-orbit coupling

3 Time reversal with spin-orbit coupling 

We now remove the restriction that // is real, introduce the symbol 0 for the time-reversal symmetry operator, and choose t0 = 0. Now Qip is the transformed function which has the 

The state described by ip(0) evolving backwards in time for the same time interval becomes one described by 

In this case (c = — 1) ( h/j and ij are orthogonal and so correspond to different degenerate states. 

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

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In the non-relativistic domain one-electron operators can be classified as triplet and singlet operators, depending on whether they contain spin operators or not. In the relativistic domain the spin-orbit interaction leads to an intimate coupling of the spin and spatial degrees of freedom, and spin symmetry is therefore lost. It can to some extent be replaced by time-reversal symmetry. We may choose the orbital basis generating the matrix of Hx to be a Kramers paired basis, that is each orbital j/p comes with the Kramers partner = generated by the action of the time-reversal operator We can then replace the summation over individual orbitals in (178) by a summation over Kramers pairs which leads to the form... 

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