Unitary transformation theorem

As discussed in detail in [10], equivalent results are not obtained with these three unitary transformations. A principal difference between the U, V, and B results is the phase of the wave function after being h ansported around a closed loop C, centered on the z axis parallel to but not in the (x, y) plane. The pertm bative wave functions obtained from U(9, <])) or B(0, <()) are, as seen from Eq. (26a) or (26c), single-valued when transported around C that is ( 3 )(r Ro) 3< (r R )) = 1, where Ro = Rn denote the beginning and end of this loop. This is a necessary condition for Berry s geometric phase theorem [22] to hold. On the other hand, the perturbative wave functions obtained from V(0, <])) in Eq. (26b) are not single valued when transported around C. 

MSN. 112. T. Petrosky and II. Prigogine, Poincare s theorem and unitary transformations for classical and quantum systems, Physica, 147A, 439 60 (1987). 

The final theorem is that a unitary transformation leaves a unitary matrix unitary. 

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a... 

Once this discussion of the space-inversion operator in the context of optically active isomers is accepted, it follows that a molecular interpretation of the optical activity equation will not be a trivial matter. This is because a molecule is conventionally defined as a dynamical system composed of a particular, finite number of electrons and nuclei it can therefore be associated with a Hamiltonian operator containing a finite number (3 M) of degrees of freedom (variables) (Sect. 2), and for such operators one has a theorem that says the Hamiltonian acts on a single, coherent Hilbert space > = 3 (9t3X)51). In more physical terms this means that all the possible excitations of the molecule can be described in . In principle therefore any superposition of states in the molecular Hilbert space is physically realizable in particular it would be legitimate to write the eigenfunctions of the usual molecular Hamiltonian, Eq. (2.14)1 3 in the form of Eq. (4.14) with suitable coefficients (C , = 0. Moreover any unitary transformation of the eigen-... 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

The preceding discussion is relevant for vertical ionization potentials only, i.e. for ionization processes without a change in the shape of the molecule. The effect of molecular distortions due to ionization has to be estimated separately and added to AEt and Et— ). Recently, the validity of Koopmans theorem has been questioned within the SCF approximation itself. It should be noted that the orbitals of a Slater determinant are defined except for a unitary transformation, and the... 

A unitary transformation of the orbitals preserves the orthogonality of the MOs. The Brillouin theorem permits one to consider not only unitary transformations, but any non-singular linear transformation, too. In the latter case the transformed MOs will not be orthogonal to each other, but Eq. (11.12) still holds. 

Our theorem permits the following inference. The statistical matrix of every pure case in quantum mechanics is equivalent to an elementary matrix and can be transformed into it by a similarity transformation. Because p is hermitian, the transforming matrix is unitary. A mixture can, therefore, always be written in the diagonal form Eq. (7-92). 

The representations Fi, F2, Ts given above are all irreducible. Since matrices representing transformations of interest to us are unitary, we may restrict ourselves to representations which involve only unitary matrices and to similarity transformations with unitary matrices. Two irreducible representations which differ only by a similarity transformation are said to be equivalent. We shall now show that the non-equivalent irreducible representations Fi, F2, F3 given above are the only non-equivalent irreducible representations of the corresponding group, and we shall then state, without proof, certain general theorems regarding irreducible representations. 

Proof. We shall prove this theorem only for the type of groups in which we have been interested, namely, those representing transformations of coordinates. The corresponding matrix representations involve only unitary matrices for simplicity, we will assume that the matrices are real. According to Theorem 1, any bilinear form... 

---