Operators under time reversal

Fig. 16. Variation in a stationary cycling state of catalyst temperature, S03, and complex concentrations in the melt phase and the concentration of gas phase species with time in a half cycle in the forward flow portion of a reactor operating under periodic reversal of flow direction with r = 40 min, SV = 900 h (Csodo = 6 vol%, (Co2)o = 15 vol%, Ta = 50°C. Curves 1, just after switching flow direction 2,1 min 3, 6.6 min 4, 13.3 min, and 5, 20 min after a switch in flow direction. (Figure adapted from Bunimovich et at., 1995, with permission, 1995 Elsevier Science Ltd.)...

Thus, real operators are unaffected by time reversal but linear and angular momenta, which have factors of i, change sign under time reversal. Therefore,... 

In this expression the F operator is broken up into its irreducible quasi-spin parts. The behavior of F under time reversal thereby proves to be the determining factor. Two cases are possible. 

The s.o.c. operator is a one-electron operator which is even under time reversal, and non-totally symmetric in spin and orbit space. The trace of the spin-orbit coupling matrix for the t2g-shell thus vanishes. As a result the s.o.c. operator is found to transform as the MK = 0 component of a pure quasi-spin triplet (Cf. Eq. 26). Application of the selection rule in Eq. 28 shows that allowed matrix elements must involve a change of one unit in quasi-spin character, i.e. AQ = 1. Since 4S and 2D are both quasi-spin singlets while 2P is a quasispin triplet, s.o.c. interactions will be as follows ... 

Use of the charge-exchange mechanism, reaction (8.13), to produce antihydrogen was first proposed by Deutch et al. (1986), and subsequently it was shown that the cross section for this process could be obtained by applying the charge conjugation and time reversal operators to the process of positronium formation in positron-hydrogen collisions (Humberston et al., 1987, and see section 4.2). Under time reversal, the positronium formation process equation (4.5) becomes... 

The diagonal matrix elements between half-filled shell states are now considered. If it is assumed that the interaction operator is symmetric under time reversal (also as in case 2), then thm = +1. The diagonal interaction elements are just the expectation value of in the closed shell, which is zero if is not totally symmetric under spatial operations (Another way of saying this is that (H%) vanishes if Mr K X K KMKr ), but obviously has time reversal parity +1. It now follows that if the above criteria are met then the diagonal matrix elements must vanish. 

The prerequisite for the creation of orientation in the aligned state can also be formulated in terms of the time reversal properties of a Hamiltonian operator which represents the perturbation. As is shown in [276, 277] the alignment-orientation conversion may only take place if the time invariant Hamiltonian is involved. For instance, the Hamiltonian operator of the linear Zeeman effect is odd under time reversal and is thus not able to effect the conversion, whilst the operator of the quadratic Stark effect is even under time reversal and, as a consequence, the quadratic Stark effect can produce alignment-orientation conversion. 

Although the Hamiltonian does not change sign under time reversal the r.h.s. must obviously do so. Rather than abandon the notion of time reversal as an invariance in quantum theory a different interpretation of the operator It may be considered. Under the usual formulation... 

The operation of time reversal interchanges the initial and final states of a colliding system. The invariance of collision amplitudes under time reversal is the principle of detailed balance. It is observed to hold for electron—atom collisions. We are interested in finding the form of the time-reversal operator 9 and its effect on electron states. 

In the absence of an external magnetic field, the Hamiltonian H is a real Operator. Then, the Schrodinger equation for an ordinary wavefunction, will be invariant under the combined operation of time reversal and complex conjugation ... 

In the course of the calculation in Section V, we used the fact that <5n (k)5 (-k)> =0, which follows because and 8n have opposite parity under time reversal. As a consequence of the different evolution operators for i>0 and r<0, the foregoing result is no longer true. Therefore we briefly sketch the modification in the results of Section V. 

In standard QM, the reversibility in time is a manifestation of a Hermifian (self-adjoint) system with stationary states and is reflected in the unitarity of the S-matrix. Unitarity entails the inclusion of the contribution of fime-reversed states. In other words, for a stationary state, invariance under time-reversal implies that if is a stationary wavefunction, then so is A major tool for deriving results in the framework of a Hermitian formalism, explicitly or implicitly, is the resolution of the identity operator, I, on the real axis, which is a Hermitian projection operator. 

The T =0 time-dependent mean-field theory currently provides the best description of nuclear dynamics at low energies [5,6]. We consider two single-particle operators, Q, P interpreted as a collective coordinate and a collective momentum. Their nature depends on the kind of motion that we want to focus upon. We require that Q Q and P-r — P under time reversal and that IP, Q] 0. We then form a constrained Hartree-Fock (CHF) calculation on the many-body Hamiltonian H by minimizing the functional... 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

For four-component wave functions it is the spin operator E that changes sign under time reversal, and by simple extension of the algebra above we can write the four-component operator as... 

Operators, Matrix Elements, and Wave Functions under Time-Reversal Symmetry... 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

The first step in the implementation of time-reversal symmetry is to classify the basic operators according to their behavior under time reversal effected by the operator K, = —iSyKo. ... 

To develop relations under time reversal, we use the creation operator a corresponding to lIt , that is... 

We should be prepared to handle operators that are antisymmetric under time reversal as well as those that are symmetric. We introduce a sign factor t to keep track of this behavior, that is... 

These operators can be made symmetric under time reversal by multiplying the E operators by i however, this makes it more difficult to define the two-particle excitation operators e. 

The case of an operator that is antisymmetric under time reversal, that is, t = -1, may now be discussed on the basis of the equations above. All we need to do is replace a by -a and b by -b. For vanishing off-diagonal elements we then get the condition... 

We now turn our attention to the entire space of ny Kramers pairs that transform under the irrep y. We see that operators are represented as 2ny x 2ny matrices. By a suitable reordering, the representation matrix for an operator symmetric under time reversal may be brought to the form... 

This form of the matrix equation displays the structure of the matrix representation of an operator that is symmetric under time reversal, given in (10.30)... 

Thus K commutes with the wave operator and the wave operator is therefore symmetric under time reversal. Because of the structure of the wave operator, each term f) must be symmetric under time reversal. We now introduce a Kramers-pair basis, and apply the time-reversal operator to the f term, to obtain... 

The only way to get a wave operator that is symmetric under time-reversal symmetry is to impose the restriction from the beginning. While this fixes the relations between the amplitudes, it also forces the occupied and the unoccupied Kramers components of the open shell to be treated equivalently. This equivalence is what introduces the ambiguity in the treatment of the open shell the open-shell Kramers pair must behave as both a particle and a hole, and the result is that the truncated commutator expansions in the coupled-cluster equations are much longer than in closed-shell theory. The alternative is to use an unrestricted wave operator with the Kramers-restricted spinors. The use of the latter provides some reduction in the work due to the relations between the integrals, but not a full reduction (Visscher et al. 1996). 

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