Projective Hilbert Spaces

It is natural to ask which operations descend from V toP(V). IsP(V) a complex vector space Usually not. If V has a complex scalar product, does P( V) have a complex scalar product No. But. as we will see in this section, a complex scalar product on V does endow P(V) with a useful notion of orthogonality. Furthermore, using the complex scalar product on V we can measure angles in P( V). At the end of the section we apply this new technology to the qubit P(C ). 

Physics books on quantum mechanics are full of expressions such as l+y) = 4= l+z + l-z)  

If the kets label individual states, i.e.. points in projective space, and if addition makes no sense in projective space, what could this addition mean The answer lies with the unitary structure (i.e., the complex scalar product) on V and how it descends to P(y). If V models a quantum mechanical system, then there is a complex scalar product ( , ) on V. Naively speaking, the complex scalar product does not descend to an operation on P(V). For example, if v, w e V 0 and v, w Q v/e have u 2v but (v, w) 2 v, w) = 2v, w). So the bracket is not well defined on equivalence classes. Still, one important consequence of the bracket survives the equivalence orthogonality. 

Note that this definition does not depend on the choice of v and w inside their equivalence classes. If v, w = 0, then for any v u and w w we have nonzero complex numbers Cv and c such that 

So it does make sense to say that two equivalence classes, i.e., two points of projective space, are orthogonal. 

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For the Berry phase, we shall quote a definition given in [164] ""The phase that can be acquired by a state moving adiabatically (slowly) around a closed path in the parameter space of the system. There is a further, somewhat more general phase, that appears in any cyclic motion, not necessarily slow in the Hilbert space, which is the Aharonov-Anandan phase [10]. Other developments and applications are abundant. An interim summai was published in 1990 [78]. A further, more up-to-date summary, especially on progress in experimental developments, is much needed. (In Section IV we list some publications that report on the experimental determinations of the Berry phase.) Regarding theoretical advances, we note (in a somewhat subjective and selective mode) some clarifications regarding parallel transport, e.g., [165], This paper discusses the projective Hilbert space and its metric (the Fubini-Study metric). The projective Hilbert space arises from the Hilbert space of the electronic manifold by the removal of the overall phase and is therefore a central geometrical concept in any treatment of the component phases, such as this chapter. 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

Fubini-Study metric, projective Hilbert space, 209 -210... 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

One-I-frame system quantum states are sustained by the whole set of interacting material elements. These states are elements of a projected Hilbert space. 

Ht-t -j acts in a projected Hilbert space without double occupancy, where... 

The question to be asked is Under what conditions (if at all) do the components of X fulfill Eq. (B.8) In [34] it is proved that this relation holds for any full Hilbert space. Here, we shall show that this relation holds also for the P sub-Hilbert space of dimension M, as defined by Eq. (10). To show that we employ, again, the Feshbach projection operator foraialism [79] [see Eqs. (11)]. 

More abstractly the condition Tr (p+p ) = iV implies that the part of the Hilbert space defined by the projection operator p should be fully contained in the part defined by the projection operator p+. If we now vary p slightly so that this condition is no longer fulfilled, Eq. II.GO shows that the pure spin state previously described by the Slater determinant becomes mixed up with states of higher quantum numbers S = m+1,. . . . The idea of the electron pairing in doubly occupied orbitals is therefore essential in the Hartree-Fock scheme in order to secure that the Slater determinant really represents a pure spin state. This means, however, that, in the calculation of the best spin orbitals y>k(x), there is a new auxiliary condition of the form... 

The von Neumann Projection Operators.—Consider the eigenstates n > in the Hilbert space of N particles with the properties ... 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

This operation uniquely determines the action of T on any vector of the Hilbert space Ln. The result can be expressed in terms of m projection operators defined on the eigenspaces Mi such that the action of Pt on u gives the projection of u on Mt, or... 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

Let be Bm = electron function basis. Slater-determinants constructed over Bm span an orthonormal, jx = ( ) dimensional subspace of the N-electron Hilbert space. The projection of the exact wave function in this subspace ( ) can be given as a linear... 

The physical idea behind the PT is the fragmentation of the Hilbert space H of the problem under study into two subspaces, usually termed Q and P, for which solutions of the corresponding SE are obtainable, while the solution of the SE for the full H is computationally formidable. By the partitioning of 7i into two subspaces, one is able to write, in each subspace independently, the SE for the projection of the unknown solution of the SE in H, while the dynamical effect of the complementary subspace is fully incorporated. In this way, one ultimately can construct the solution of the SE in H after solving for its projections in Q and P independently. 

The starting point for Lowdin s PT [1-6] and Eeshbach s projection formalism [7-9] is the fragmentation of the Hilbert space H = Q V, of a given time-independent Hamiltonian H, into subspaces Q and V by the action of projection operators Q and P, respectively. The projection operators satisfy the following conditions ... 

Now we will briefly indicate the problems that can be usefully treated with the above geometric theory of the hydrogen atom." In many applications, such as the theory of the Compton effect in a bound electron and in the inelastic matter theory of atoms it is a question of determining the norm of the projection of a given function on the subspace of Hilbert space determined by the principal quanmni number nJ This norm is defined by the sum... 

Here P = 1 — IfoX ol is the projection operator for the part of the Hilbert space which is the orthogonal complement to [Pg.4]

At the Fence space in which laboratory and Hilbert space magnitudes are gathered, one gets projected amplitude ... 

The I-frame localizes a projected quantum state this frame is a classical physics element, whereas configuration base states x) define a rigged Hilbert space basis [1]. 

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