Eigenvector phases

Now, because eigenvector phases are arbitrary, one can choose all the a/o to be positive. Since one knows the aftt, then by this choice of phase the a,tt are known as well. 

Pauli spin matrices, geometric phase theory, eigenvector evolution, 14-17... 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

The Hamiltonian will in general be a function of time, but the special case of a time-independent Hamiltonian is of central importance. Suppose that H has an eigenvalue of E and an eigenvector v at t = 0. Suppose that the state of the system at any other time t is u(t) = wexp(—iat), where the argument of the phase factor is to be established. The derivative... 

According to the well-known Landau theory, the eigenvector of the order parameter in any second order solid-solid phase transition transforms according to an irreducible representation of the space group of the parent phase state. Furthermore, the free energy F=U -TS can be expanded around the transition temperature Tc in terms of the scalar order parameter p, which... 

The soft mode concept can be extended to all distortive phase transitions (transitions with relatively small atomic displacements), even if they are only close to second order. In the case of a ferro-distortive transition, as for example in BaTiOs or KDP, the order parameter is proportional to the spontaneous electric polarization Fj. d F/ dp is not only proportional to co, but also to the dielectric susceptibility. This does not, however, mean that all components of the order parameter eigenvector must contribute to Ps. 

In multiscale asymptotic analysis of reaction network we found several very attractive zero-one laws. First of all, components eigenvectors are close to 0 or +1. This law together with two other zero-one laws are discussed in Section 6 "Three zero-one laws and nonequilibrium phase transitions in multiscale systems". 

Consider the case that the columns of U t) are the eigenvectors of H(t). Then U(t)H t)U t) is diagonal and the adiabatic approximation asserts that if U t)H t)U t) the time evolution of an initial superposition of eigenstates of H t), denoted k)(, remains unchanged aside from the phases... 

Next we turn to the estimation of the eigenvectors. Let normalized eigenvectors of R associated with the eigenvalues Zj% (j— 1,. .., m). phase factor since are different from each other by (10. 7). Noting that (R% — Xv. ) wc have... 

As discussed in Section 10.3, equilibrium phase homogeneity is associated with existence of a null eigenvector t] of the full (c + 2)-dimensional metric matrix... 

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