Direct-product space

States can be simultaneously eigenstates of more than one observable. In this case the observables commute. If the simultaneous eigenstates of a set of commuting observables form a complete set they span a new space which is the direct product space of the spaces spanned by the eigenstates of each of the observables in the set. The dimension of the new space is the product of the dimensions of the spaces spanned by the eigenstates of the individual observables. 

The computational procedure involves obtaining the matrix representation of the symmetry operators of the (2n + 2) site chain in the direct product basis. The matrix representation of both J and P for the new sites in the Fock space is known from their definitions. Similarly, the matrix representations of the operators J and P for the left (right) part of the system at the first iteration are also known in the basis of the corresponding Fock space states. These are then transformed to the density matrix eigenvectors bcisis. The matrix representation of the symmetry operators of the full system in the direct product space are obtained as the direct product of the corresponding matrices ... 

The trace over a direct product space is the product of the traces of the factor spaces, such that... 

We can compute the actual relative intensities of transitions between atomic states from previously developed symmetry considerations. Suppose an integrand contains the product of three functions u, r, and v each of which belongs to a set of base vectors for a representation of C(3). We may choose two of the functions, say v and r, and express their product in terms of new base vectors which span the direct product space r(v) X r(r). As an example of this procedure we will compute the relative probabilities for the transition from the states Ji I, Mj, = 0, I to the states J2 = I, = 0, 1. Such transitions are typically... 

Just as the use of a finite basis set in independent-electron models restricts the domain of the relevant one-electron operator, h, so the algebraic approximation results in the restriction of the domain of the total Hamiltonian to a finite-dimensional subspace of the Hilbert space. In most applications of quantum mechanics to atoms and molecules which go beyond the independent-electron models, the JV-electron wavefunction is expressed in terms of the fVth-rank direct product space M generated by a finitedimensional single-particle space MlS that is... 

In the case of N Cgo Sn = i/2 and = 1/2 spanning an eight-dimensional direct product space. The total spin can have the values of 5 = 1 or 5 = 2 with the corresponding energies... 

Let us now consider the compound system. We assume that the compound wave function is written as a linear combination of determinants in the direct-product space of the fragment spaces... 

The compound wave function is thus determined in exactly the same manner as the fragment wave functions but in the direct-product space of the fragment spaces. We shall now demonstrate thaL for noninteracting systems A and B, the variationally optimized energy of the compound system in the direct-product space is equal to the sum of the energies of the fragments... 

Clearly, we cannot generate all possible wave functions from (4.3.52) since this would require the use of the full direct-product space as previously discussed. On the other hand, the exponential ansatz, with operators and parameters referring only to the individual subsystems, provides exactly those operators that are needed to describe the products of isolated systems. Assume, for example, that some choice of operators Xia and X,b has been made and that the parameters Pa have been cbosen to optimize the energy of fragment A... 

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

We then dehne an internal coordinate <() such that <() = 0 2ti denotes a a path that has described one complete loop around the Cl in the nuclear branching space. Other than this, we need specify no further details about < ). We do not even need to specify whether the complete set of nuclear coordinates give a direct product representation of the space. It is sufficient that closed loop has wound around the CL Using this definition of ((), we can express the effect of the GP on the... 

Clearly, the above procedure can be continued (in principle) as many times as required. Thus, if the wave function includes n = —4 3 paths, we have simply to dehne the function I 4((t)) = —+ 8ti), and then map onto the (j) = 0 16ti cover space, which will unwind the function completely. In general, if there are h homotopy classes of Feynman paths that contribute to the Kernel, then one can unwind ihG by computing the unsymmetrised wave function ih in the 0 2hn cover space. The symmetry group of the latter will be a direct product of the symmetry group in the single space and the group... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

Vector spaces which occur in physical applications are often direct products of smaller vector spaces that correspond to different degrees of freedom of the physical system (e.g. translations and rotations of a rigid body, or orbital and spin motion of a particle such as an electron). The characterization of such a situation depends on the relationship between the representations of a symmetry group realized on the product space and those defined on the component spaces. 

The group (E, J) has only two one-dimensional irreducible representations. The representations of 0/(3) can therefore be obtained from those of 0(3) as direct products. The group 0/(3) is called the three-dimensional rotation-inversion group. It is isomorphic with the crystallographic space group Pi. 

Direct Product Table for C2v Point Group in Coordinate Space with / to Give the D2h Point Group in Momentum Space... 

Here we find a new concept, the direct product between irreducible representations of a symmetry group. This direct product is related to the product of their corresponding space functions. For our purposes, we will only mention that the direct product between two, Pj and A, (or more) irreducible representations of a group is a new... 

The calculated state energies, the transition moments, and the symmetry classification are given in Table 3. The symmetry species of the triplet functions is obtained by taking the direct product of irreducible representation of the space and the spin functions Fx, Fy, Fz, which transform as the rotations Rx, Ry, and Rz-... 

In order to evaluate the spectral density of Eq. (35) or (38), one needs a complete basis set spanning the lattice operator space. This basis set can be obtained by taking direct products of Wigner rotation matrices,... 

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