Unitary operator

To consider that it is and not the form of each individual m.o. that ultimately matters is a consequence of the determinantal form of the wavefunc-tion. In fact, determinants remain unchanged when the respective elements are subject to some specific operations (unitary transformations) as was already illustrated on page 89 (Problem 5.2). For example, the eigenfunction (Eq. (8.15)) is not changed if we replace each m.o. (row i) by Xi= + Such mathematical alterations inside the determinant do not correspond to any physical change, because the eigenfunction S remains unchanged. 

The appropriate quantum mechanical operator fomi of the phase has been the subject of numerous efforts. At present, one can only speak of the best approximate operator, and this also is the subject of debate. A personal historical account by Nieto of various operator definitions for the phase (and of its probability distribution) is in [27] and in companion articles, for example, [130-132] and others, that have appeared in Volume 48 of Physica Scripta T (1993), which is devoted to this subject. (For an introduction to the unitarity requirements placed on a phase operator, one can refer to [133]). In 1927, Dirac proposed a quantum mechanical operator tf), defined in terms of the creation and destruction operators [134], but London [135] showed that this is not Hermitean. (A further source is [136].) Another candidate, e is not unitary. 

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... 

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

A unitary transformation is then introduced whieh diagonalizes the FG matrix, yielding eigenvalues s, and eigenvectors q,. The kinetic energy operator is still diagonal in these eoordinates. 

This formula can be used directly, but to compare with more standard perturbation theory, we can use, for instance, the unitary operator (Kato 1980, p.99)... 

Although the two quantum models are defined somewhat dilferently, both QCA-I and QCA-II start with the same basic premise, endowing the classical system with two characteristically quantum features. They both (1) replace each site variable with a quantum state containing all fc classical site-color possibilities, and (2) introduce a quantum transition operator "I , defining mixed color —> mixed a)lor transitions. Only in QCA-II, however, is also unitary see discussion below. 

Focusing on strictly local (i.e. nearest neighbor) interactions. Grossing and Zeilinger [gross88a] consider the following unitary evolution operator U, approximated to first order by ... 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

The problem now is to find the corresponding Hamiltonian, t Hooft shows that the most obvious construction, obtained by rewriting U(t+l,t) as a product of cyclic elements, unfortunately does not work because at the end of the calculation there is no way to uniquely define the vacuum state. Given a cellular automaton with a local unitary evolution operator U = WgUg and the commutator [Ug, Ug ] 0 if [ af — af j> d for some d > 0, the real problem is therefore to find a Hamiltonian... 

The operator D( ) is, therefore, unitary, since the components of P are hermitian (and commute, for otherwise a formula like (7-4) would have no meaning). 

Rotations are likewise unitary transformations, and we shall see that they can also be represented by an exponential operator. Let D(a) be a rotation about the z-axis, so that... 

Note that we are not transforming the vector operator itself—its form is independent of rotation. Hence, Eq. (7-11) must be satisfied by subjecting the states ijt to a unitary transformation U ... 

In this equation we transform both tf> and tft by a unitary operator U, which represents a rotation whose matrix is R (R acts on c), obtaining... 

Now let us use the set, <0> to form a matrix representation of some operator Q at time hi assuming that Q is not explicitly a function of time. The expectation value of Q in the various states, changes in time only by virtue of the time-dependence of the state vectors used in the representation. However, because this dependence is equivalent to a unitary transformation, the matrix at time t is derived from the matrix at time t0 by such a unitary transformation, and we know that this cannot change the trace of the matrix. Thus if Q — WXR our result entails that it is not possible to change the ensemble average of R, which is just the trace of Q. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

Equivalently, this unitary operator S can be defined by the relation... 

If the hamiltonians H(0) and H0(0) are such that there exist no bound states, and the states F )+ are properly normalized, l(+) is a unitary operator. Furthermore, it has the property that... 

The representation of these commutation rules is again fixed by the requirement that there exist no-particle states 0>out and 0>ln. The -matrix is defined as the unitary operator which relates the in and out fields ... 

Such a unitary operator 8 must exist since tfiia and rout (and s inil and form equivalent representation of the commutation rules (11-60) and (11-65). Explicitly it can be computed as follows... 

In the Heisenberg-type description the existence of such a unitary or anti-unitary operator U is inferred from the fact that the set of observable Q and Q satisfy the same commutation rules. 

Let us next adopt the Schrodinger-type description. The statement that quantum electrodynamics is invariant under space inversion can now be translated into the statement that there exists a unitary operator U(it) such that... 

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