Irreducible representations invariant operators

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... 

Invariance principle, 664 Invariance properties of quantum electrodynamics, 664 Inventory problem, 252,281,286 Inverse collisions, 11 direct and, 12 Inverse operator, 688 Investment problem, 286 Irreducible representations of crystallographic point groups, 726 Isoperimetric problems, 305 Iteration for the inverse, 60... 

Corresponding relations for arbitrary space group elements show that if the orbitals r) which make up the density transform asthe irreducible representations of the space group, the density is invariant under all the operations of that group. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

A further property of die dieter tables arises from the fact that every symmetry group has an irreducible representation that is invariant under all of die group operations. This irreducible representation is a one-by-one unit matrix (the number one) for every class of operation. Obviously, the characters, are all then equal to one. AS this irreducible representation is by convention taken to be the first row of all Character tables consists solely of ones. The significance of the character tables will become more apparent by consideration of an example. 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

An important case is where F in (9.183) is replaced by the Hamiltonian operator H. The Hamiltonian is invariant under all symmetry operations, so that the function belongs to the irreducible representation Tp. For... 

Now when we say that the integrand fAfB is invariant to all symmetry operations, this means that it forms a basis for the totally symmetric representation. But from what has been said above, we know how to determine the irreducible representations occurring in the representation for which forms a basis if we know the irreducible representations for which fA and fB separately form bases. In general ... 

The coefficients a must be so chosen that 2 0,- transforms in the fashion appropriate to the irreducible representation r. Now the important point is that bases for only certain irreducible representations can be constructed out of linear combinations of the vL- To determine which, one ascertains the group characters associated with the transformation scheme, usually reducible, of the original attached wave functions rpi before linear combinations are taken. This step is easy, as the character xd for a covering operation D is simply equal to q, where q is the number of atoms left invariant by D. This result is true inasmuch as D leaves q of the atoms alone, and completely rearranges the others, so that the diagonal sum involved in the character will contain unity q times, and will have zeros for the other entries. The scheme for evaluating the characters is reminiscent of that in the group... 

Here x(C4), for instance, means the character for the covering operation consisting of rotation about one of the fourfold or principal cubic axes (normals to cube faces) by 2a-/4. Any rotation about such an axis leaves two atoms invariant, and hence x(Cs) = x(C4) = 2- On the other hand, x((Y)=x(Q)—0 since no atoms are left invariant under rotations about the twofold or secondary cubic axes (surface diagonals) or about the threefold axes (body diagonals). Inversion in the center of symmetry is denoted by I. By using tables of characters for the group Oh, one finds that the irreducible representations contained in the character scheme (2) are, in MuIIiken s notation,4... 

After the factorization, one can derive SALCs within each invariant subspace via the action of the projection operator. The projection operator for irreducible representation j is defined as... 

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

The expression for the intermolecular potential must satisfy two symmetry requirements. First, it must be invariant if we rotate the molecular frame of either of the two molecules through specific Euler angles wg that correspond with a symmetry element of the molecule in question. This means that our basis must be invariant under rotations of the outer direct product group Gp GP-, where GP is the symmetry group of molecule P and GP- that of molecule F Acting with the projection operator of the totally symmetric irreducible representation of the group GP (of order GP),... 

Consequently the spinors < that make up Q must transform according to the irreducible representations or corepresentations (if the invariance group contains antiunitary operators) of the invariance group. 

Weyl answered the first point with the insight that all irreducible representations of the special linear group can be made as invariant subspaces of tensor powers of the underlying standard representation. They were conceived as operations of the linear group transformations with a determinant on a geometrical coordinate space. Any representation of the linear group can be characterized with a tensor product of the coordinate space by a symmetry property. 

Since the operator YSn m nm is invariant upon the crystal space group, and the functions ( (k) and I g fk) transform upon irreducible representation of the space group, the matrix element (3.113) will not be zero only for excitonic states (p(/),k) and (p( ),k) which transform upon the same irreducible representation of the crystal group. 

That all the operators in are invariant under the transformation t is obvious except perhaps for which depends on the orbitals. Vf, Eq. (8), can be written in terms of the Fock-Dirac density p(x<,X/). The summation in p(X<,x ) covers a set of molecular orbitals which form a basis for certain irreducible representations. This is invariant under any unitary transformation. ... 

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