Direct tensorial product

It is clear then that one needs to describe all relevant operators (related to the spin components) using their matrix representations in the four-fold vector space generated by the state vectors mj, ms). These matrices can be constructed by evaluating each matrix element or they can be built by the direct tensorial product of the corresponding 2x2 matrices that describe the dynamics of the separate spin 1 /2 systems [5,12], The direct tensorial product of two nxn matrices A and B lead to a x matrix C = A B whose elements are... 

Using second-quantization, it is often necessary to transform complicated tensorial products of creation and annihilation operators. If, to this end, conventional anticommutation relations (14.19) are used, then one proceeds as follows write the irreducible tensorial products in explicit form in terms of the sum over the projection parameters of conventional products of creation and annihilation operators, then place these operators in the required order, and finally sum the resultant expression again over the projection parameters. On the other hand, the use of (14.21) enables the irreducible tensorial products of second-quantization operators to be transformed directly. 

Let us now return to the Casimir operators for groups Spy+2, SU21+1, R21+1, which can also be expressed in terms of linear combinations of irreducible tensorial products of triple tensors WiKkK To this end, we insert into the scalar products of operators Uk (or Vkl), their expressions in terms of triple tensors (15.60) and then expand the direct product in terms of irreducible components in quasispin space. As a result, we arrive at... 

For the p-shell we do not need these additional relationships, since from the set of equations (15.71)—(15.74), (15.82) and (15.83) we can work out expressions for all the irreducible tensorial products. Directly from (15.71) we obtain (hereinafter a letter over the equality sign shows what shell this particular equation is valid for)... 

The energy operator of electrostatic interaction has the same expression as in the non-relativistic case (H = Q, where Q is defined by (1.15)). Its irreducible form is given by (19.6). In order to find the irreducible form of the operator of magnetic interactions H-, we make use of expansion (19.6) and then transform the coupling scheme of tensors to one in which the operators acting on one and the same coordinates, would be directly coupled into a tensorial product. This gives finally... 

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... 

It is valuable to consider the direct product of the two sets i / and 2 / which form the bases for the same irreducible representation of i 3. One obtains in the direct sum symmetrical and skew-symmetrical irreducible tensorial sets... 

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