Product Hilbert space

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

For a density matrix pab defined on a tensor producted Hilbert space Ha 2 Ttb this state is unentangled if it may be written as a convex sum of unentangled states, namely... 

When we switch to the quantum mechanical treatment in the next section the electronic and nuclear densities will be expressed by the corresponding wave functions and the tensor contraction finds its counterpart in the formalism of product Hilbert spaces. 

Let the bits in the memory of the quantum computer be represented by a chain of three spins. Then each state of our computer is described by a 2 -dimensional vector. U(l) becomes a 2 x 2 matrix. It acts on the tensor product Hilbert space of the spaces corresponding to the three spins. We write 0) = [J] for a... 

Composite systems. If two quantum systems are described by two Hilbert spaces TLoo and the combined system is described by the product Hilbert space = TLoo In li we can distinguish between product states and entangled states. The former can be written as a product of a state in T-Lqq and a state 5 2) in e as ) = For entangled states this is not possible. We will... 

A system of two qubits is described by the product Hilbert space An... 

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

Note that relations (1.91) and (1.92) mean linearity of the duality mapping I and its inverse I in Hilbert spaces due to the linearity of the scalar product. 

On the other hand, the duality mapping I is defined by the scalar product in the Hilbert space V. Assume that the operator A is self-conjugate. Then we can define the scalar product in V as follows ... 

Postulate B.—There exists a set of vectors indicated by the bra symbol < in one-to-one correspondence with the vectors of 3/f, forming a dual Hilbert space 3 . As a matter of notation we use the symbol , etc. This dual space must be such that a meaning can be given to the scalar product of any vector with the following properties ... 

Other postulates required to complete the definition of will not be listed here they are concerned with the existence of a basis set of vectors and we shall discuss that question in some detail in the next section. For the present we may summarize the above defining properties of Hilbert space by saying that it is a linear space with a complex-valued scalar product. 

The fact that we are discussing an abstract space means that we know only that its elements (vectors) have the postulated properties e.g., that a scalar product exists, but at this level of the discussion we do not know the numerical value of the scalar product. We may choose at random some familiar collection of elements, perhaps the set of all ordered pairs of real numbers (n,m) or the set of all differentiable functions of position on a line, etc., and ask whether or not they form a Hilbert space. If they do, then we can in fact evaluate the scalar... 

The quantum states of Schrodinger s theory constitute an example of Hilbert space, and their scalar product has a direct physical meaning. Any such example of Hilbert space, where we can actually evaluate the scalar products numerically, is called a representation of Hilbert space. We shall continue discussing the properties of abstract Hilbert space, so that all our conclusions will apply to any and every representation. 

It will be noticed that continuous basis sets, with improper Dirao delta functions as scalar products, do not strictly belong to Hilbert space as defined in Section 8.3, where the basis is specifically required by postulate to be denumerably infinite. The nondenumerably infinite sets g> or j actually span what is known as Banach spaces,5 but we shall here conform to the custom among theoretical physicists to oall them Hilbert spaces. 

The solutions of Eq. (9-63) form a Hilbert space when the scalar product is defined by... 

A Lorentz invariant scalar product can be defined in the linear vector space formed by the positive energy solutions which makes this vector space into a Hilbert space. For two positive energy Klein-... 

The indefinite bilinear form (XF, TB) as defined by Eq. (9-687) can be expressed in terms of the Hilbert space scalar product (Y,x)h 85 follows ... 

In effect the scalar product in (9-688), which makes the vector space into a Hilbert space, omits the factor ( —1) from the bilinear form (9-687). We shall always work with the indefinite bilinear form (9-687). Thus, for example, one verifies that with this indefinite metric... 

The complete set of states obtained by applying products of and on lO) also spans the Hilbert space of physical states. These out states are specified in terms of measurements performed at time = +oo, i.e., in the remote future. 

Transformations in Hilbert space, 433 Transition probabilities of negatons in, external fields, 626 Transport theory, 1 Transportation problems, 261,296 Transversal amplitude, 552 Transversal vector, 554 Transverse gauge, 643 Triangular factorization, 65 Tridiagonal form, 73 Triple product ensemble, 218 Truncation error, 52 Truncation of differential equations/ 388... 

The Hilbert space of pure A -particle fermion states. It is an iV-foId antisymmetric tensor product of the Hilbert space of pure one-particle states. 

This set forms a Hilbert space with an inner product defined by... 

Linear bounded operators in a real Hilbert space. Let H he a real Hilbert space equipped with an inner product x,y) and associated norm II X II = (x, x). We consider bounded linear operators defined on the space... 

Let A be a positive self-adjoint linear operator. By introducing on the space H the inner product x,y) = Ax,y) and the associated norm x) we obtain a Hilbert space Ha, which is usually called the energetic space Ha- It is easy to show that the inner product... 

Let T be a self-adjoint positive definite linear operator in Hilbert space H equipped with an inner product (,) and let / be a given element of the space H. The problem of minimizing the functional... 

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 view of the preceding considerations it should be emphasized that it is incorrect to talk about the self-consistent-field molecular orbitals of a molecular system in the Hartree-Fock approximation. The correct point of view is to associate the molecular orbital wavefunction of Eq. (1) with the N-dimen-sional linear Hilbert space spanned by the orbitals t/2,... uN any set of N linearly independent functions in this space can be used as molecular orbitals for forming the antisymmetrized product. 

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