Tensor products

In the mixed quantum-classical molecular dynamics (QCMD) model (see [11, 9, 2, 3, 5] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. The full quantum system is first separated via a tensor product ansatz. The evolution of each part is then modeled either classically or quan-tally. This leads to a coupled system of Newtonian and Schrbdinger equations. 

A very convenient indirect procedure for the derivation of shape functions in rectangular elements is to use the tensor products of one-dimensional interpolation functions. This can be readily explained considering the four-node rectangular element shown in Figure 2.8. 

A similar procedure is used to generate tensor-product three-dimensional elements, such as the 27-node tri-quadratic element. The shape functions in two-or three-dimensional tensor product elements are always incomplete polynomials. 

Using this coordinate system the shape functions for the first two members of the tensor product Lagrange element family are expressed as... 

The global state of the system at any time H E(t) >, is the tensor product of the individual site states, and is therefore a vector in a fc -dimensional tensor product space C ... 

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. 

In the atomic case, the pwc s are defined by the ion level I and the / value of the electron partial wave, i.e. the formal pwc subsets span the tensor product of the ion level states times the one-electron /-wave manifold. In the following, the subspaces will be indexed with greek letters a subspace index a=/ ,/ will designate explicitly an open pwc subspace, while an index 0 an arbitrary subspace. The Ic subspace will be numbered 0 and Qp will denote the projector in the subspace 0. 

The unknown permeability is represented using tensor product B-splines, which are given by the product of univariate B-splines ... 

Quantum mechanically the combination of two angular momenta is more complicated since the angular momenta are operators. (They are tensor operators of rank one.) The law of combination of angular momenta can be expressed, in general, by the so-called tensor product of two operators, indicated by the symbol x,... 

The algebra of U(4) can be written in terms of spherical tensors as in Table 2.1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. (1.25). Note that each tensor operator of multipolarity X has 2X+ 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form. 

With the representations of Section A.8 one can form tensor products. Tensor products are usually denoted by the symbol ,... 

The basic texts on tensor products are de Shalit and Talmi (1963) and Fano and Racah (1959). 

In this picture, the correspondence between irreducible representations of F (except the trivial representation) and irreducible components of the exceptional set becomes concrete. It is realized by the tautological bundles V s. In [66, 5.8], we have shown the correspondence respects the multiplicative structures, one given by the tensor product and one given by the cup product. In fact, using (4.11), we can show that two matrices... 

Subadditivity The entanglement of the tensor product of two states should not be larger that the sum of the entanglement of each of the states. 

A very simple ansatz would be to approximate the high-dimensional coefficient tensor T by a tensor product of vectors. In elemental form, this would be... 

As in a Cartesian coordinate system the tensor product u v, of the vectors u and v, has the elements uxvp, the EFG tensor is a symmetric tensor with elements... 

The electron-phonon operator is a tensor product between the electronic dipole and the nuclear dipole operators. A mixing between the AA and BB via the singlet-spin diradical AB state is possible now. A linear superposition of identical vibration states in AA and BB is performed by the excited state diradical AB. If the system started at cis state, after coupling may decohere by emission of a vibration photon in the trans state furthermore, relaxation to the trans... 

A note on terminology operations that survive the equivalence are sometimes called well defined on equivalence classes. A function on the original set S taking the same value on every element of an equivalence class is called an invariant of the equivalence relation. We will see an example of an invariant of an equivalence class in our introduction to tensor products in Section 2.6. 

Not only are linear transformations necessary for the very definition of a representation in Chapter 6, but they are useful in calculating dimensions of vector spaces — see Proposition 2.5. Linear transformations are at the heart of homomorphisms of representations and many other constructions. We will often appeal to the propositions in this section as we construct linear transformations. For example, we will use Proposition 2.4 in Section 5.3 to define the tensor product of representations. 

---