Hilbert space subspace

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

Figure 1. Orthogonal decomposition of a three-dimensional Hilbert space geometrical representation of the two orthogonal subspaces. 

Our first way of answering the last question will be based on the fundamental theorems on Hilbert space [14], Indeed, the theorem on separability tells us that any subspace of h is also a separable Hilbert space. As a consequence, the inner product defined on, say, the occupied subspace is hermitian irrespectively of the choice of the basis x f (/)], as long as this latter satisfies the fundamental requirements of Quantum Mechanics. One should therefore not have to impose this property as a constraint when counting the number of conditions arising from the constraint CC+ =1 but, on the contrary, can take it for granted. 

In this limit, therefore, the different ground states generate separate Hilbert spaces, and transitions among them are forbidden. A superselection rule [128] is said to insulate one ground state from another. A superselection rule operates between subspaces if there are neither spontaneous transitions between their state vectors and if, in addition, there are no measurable quantities with finite matrix elements between their state vectors. 

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... 

Consider, furthermore, a (2i- - 1)-dimensional subspace of the Hilbert space with fixed 5. Then, according to Schwinger s theory of angular momentum [98], this discrete spin DoF can be represented by two bosonic oscillators described by creation and annihilation operators with commutation relations... 

The mapping preserves the commutation relations (70) of the spin operators. As can be seen from (75d), the image of the 2s + 1)-dimensional spin Hilbert space is the subspace of the two-oscillator Hilbert space with 2s quanmm of excitation— the so-called physical subspace [218, 220], This subspace is invariant under the action of any operator which results by the mapping (75a)-(75d) from an arbitrary spin operator A(5 i, S2, S3). Thus, starting in this subspace the system will always remain in it. As a consequence, the mapping yields the following identity for the matrix elements of an operator A ... 

The image of the A-level Hilbert space is the subspace of the A-oscUlator Hilbert space with a single quantum excitation. Again, this (physical) subspace is invariant under the action of any operator which results by the mapping (80a) from an arbitrary A-level system operator. As a consequence we obtain the propagator identity... 

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 ... 

One considers a Hilbert space that can be fragmented as the direct sum of three subspaces H = Qi Q2 The fragmentation of the Hilbert space in three... 

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... 

When there are more macroscopic observables B,C,... the process can be continued. The end result is a collection of coarse-grained observables //, A, 5, C, ..., which all commute with one another. The Hilbert space H is decomposed in linear subspaces that are common eigenspaces of these observables. We shall call these subspaces phase cells and indicate them with a single label J. They correspond to definite values of the coarse-grained variables, which we shall now denote by Ej, AJy BJy Cj,. These phase cells are the macrostates. 

Thus the eigenstates of HSF are labeled by the partitions [ASP] associated with the irreducible representations of S%v. These [Asp] labels are called permutation quantum numbers. The spin-free Hilbert space Fsp of the Hamiltonian may be decomposed into subspaces FSF([ASF]) invariant to 5 p... 

The Pauli-allowed portion W of the total Hilbert space of H(Qeq) may then be decomposed into subspaces W([Pg.27]

The operator (1 — Pt)L is Hermitian in the subspace of Hilbert space which is orthogonal to 17,>. To see this define the vectors... 

Needless to say, the so-called combination transitions are also considered in this subspace. Lineshape equations for special forms of the relaxation matrix can also be written in terms of the Hilbert space. However, the notation becomes quite involved. This is probably the source of some erroneous simplifications which consist of neglecting combination transitions in the equations of lineshape. (50)... 

To summarize, we have considered a quantum mechanical N-body system with dilation analytic potentials, Ey, and its dependence on the scaling parameter i] = t] (for some 0 < < 0, depending on V). To be more detailed, we need to restrict Hilbert space to a dense subspace [54], the so-called Nelson class N, which provides the domain over which the unbounded complex scaling is well-defined. Closing the subset < - D T) in ft, see [9] and references therein for a more detailed expose, one obtains the scaled version of the original partial differential equation... 

The real Hilbert space is always partitioned into a direct sum of subspaces, each representing a different energy eigenvalue of the spectrum of the hamiltonian operator ... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

In addition, the functions q, if define what is called a g,-fold subspace of total Hilbert space. Furthermore, the eigenfunctions of two states with different values of q must be orthogonal since X is a Hermitian operator. Once again we normalise these functions to unity so that (7.4) may be generalised to... 

Thus the members of the orthonormal set q, if are all orthogonal to those of the orthonormal set q, j) 0 for q f q correspondingly the subspaces defined by these two sets are said to be orthogonal to each other. For example, all the subspaces defined by q,i)° with q / 0 (i.e. excited vibronic states) are orthogonal to the subspace of the ground vibronic state. The sum total of all these subspaces with q / 0 (itself a subspace of total Hilbert space) is called the subspace complementary to the subspace of the ground vibronic state. 

The problem treated in this paper is complicated by the fact that the linear operator T is supposed to be linearly defined on a linear space A = F of all complex functions F = F(X) of a real composite variable X = jcl5 x2, , xN, of which the L2 Hilbert space is only a small subspace. We will refer to this space A = F as the definition space of the operator T. Since it contains all complex functions, it is stable under complex conjugation, and, according to Eq. (1.50) one has T F = (TF ), which means that also the complex conjugate operator T is defined on this space. 

The L2 Hilbert space T is a subspace of the linear space A, and we note that the operator V transforms the elements P(Z) into complex functions of the composite complex variable Z, so that... 

The algebraic approximation results in the restriction of the domain of the operator to a finite dimensional subspace, Sf, of the Hilbert space The algebraic approximation may be implemented by defining a suitable orthonormal basis set of M (electron spin orbitals and constructing all unique iV-electron determinants /t> using the M one-electron functions. The... 

The outline of the review is as follows in the next section (Sect. 2) we introduce the basic ideas of effective Hamiltonian theory based on the use of projection operators. The effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps first by analogy with r-electron theory we use the group product function method of Lykos and Parr to define a set of n-electron wavefimctions which define a subspace of the full -particle Hilbert space in which we can give a detailed analysis of the Schrodinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This subspace consists of fully antisymmetrized product wavefimctions composed of a fixed ground state wavefunction, for the electrons in the molecule other than the electrons which are placed in states, constructed out of pure d-orbitals on the... 

The renormalized theory of the effective Hamiltonian implied by the restriction to some subspace S of the full Hilbert space also imposes a requirement for renonnalisation of expectation values of other operators (Freed ). Suppose that we have some operator B and we require its expectation value in a state 0 of the full Schrddinger Eq. (2-2) in complete analogy with the effective Hamiltonian theory described above we define an effective operator B by the requirement that its expectation value in a state A ) in the subspace should equal the exact expectation value (c.f. Eq. (2-4)),... 

This rdation shows how the action of the antisymmetiizer can mix different orders in perturbation theory. Secondly, the projected functions Ag 0 > 0 > do not form an orthogonal set in the antisymmetric subspace of the Hilbert space L2(r3nj. jf excited states (a > and b > in order to obtain a complete... 

An effective method to acetderato the convttrgerice of the MRM algorithm is based on the Krylov-subspace method (Kleinman and van den Berg, 1993). VVe introduced the Krylov subspace in Chapter 2 as the finite dirncnsiorial subsjtace A, of the Hilbert space M, spanned by the vec.tors r , Lr . lAr,.. 

We can discretize the model, in this case the anomalous conductivity distribution, Ad (r), by introducing a set of basis functions, V l (r), V 2 (r), / /v (r) in the finite dimensional Hilbert space M, which is a subspace of the complex Hilbert space Mo M/v C Mo- Let us approximate the anomalous conductivity by its projection over the basis functions ... 

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