Eigenvalue equations systems

As has been shown previously [243], both sets can be described by eigenvalue equations, but for the set 2 it is more direct to work with projectors Pr taking the values 1 or 0. Let us consider a class of functions/(x), describing the state of the system or a process, such that (for reasons rooted in physics)/(x) should vanish for X D (i.e., for supp/(x) = D, where D can be an arbifiary domain and x represents a set of variables). If Pro(x) is the projector onto the domain D, which equals 1 for x G D and 0 for x D, then all functions having this state property obey an equation of restriction [244] ... 

This equation is an eigenvalue equation for the energy or Hamiltonian operator its eigenvalues provide the energy levels of the system... 

If the matrix A is positive definite, i.e. it is symmetric and has positive eigenvalues, the solution of the linear equation system is equivalent to the minimization of the bilinear form given in Eq. (64). One of the best established methods for the solution of minimization problems is the method of steepest descent. The term steepest descent alludes to a picture where the cost function F is visualized as a land-... 

The important states for atomic systems are those of definite energy E that satisfy the eigenvalue equation (11), which implies that the time-dependence of k for such states is given by... 

Although in principle one could choose a set of arbitrary values for the solvent coordinates sm, solve the eigenvalue equation (2.23), and compute the free energy (2.12), in practice a preliminary aquaintance with the equilibrium solvation picture for the target reaction system serves as a computationally convenient doorway for the calculations in the nonequilibrium solvation regime. We show this below in the section dedicated to an illustration of the method for a two state case reported in BH-II. 

The solution of the eigenvalue equation has automatically given us the desired decomposition of the motion of this simple system. 

Solution. First transform the problem to the principal axes system. The eigenvalue equation for D is... 

The key eigenvalue equation in chemistry is the Schrodinger equation, Hip = Eip. The solution of this equation for a particular system (such as an electron bound by the field of a nucleus) yields so called wavefunctions, ip, that completely describe the system of interest and from which any property of the system can be extracted. 

We want an eigenvalue equation because (cf. Section 4.3.4) we hope to be able to use the matrix form of a series of such equations to invoke matrix diagonalization to get eigenvalues and eigenvectors. Equation (5.35) is not quite an eigenvalue equation, because it is not of the form operation on function = k x function, but rather operation on function = sum of (k x functions). However, by transforming the molecular orbitals to a new set the equation can be put in eigenvalue form (with a caveat, as we shall see). Equation 5.35 represents a system of equations... 

The equation is assumed valid for any system. It turns into an eigenvalue equation for the total energy once the potential energy has been correctly... 

Indeed, one can analyze In the same manner the evolution of the system under consideration under conditions of reversibility of all of the elementary reactions in scheme (3.30). Unfortunately, in this situation the analytic solution of the eigenvalue equation in respect to parameter X appears unreasonably awkward. However, if the kinetic irreversibility of both nonlinear steps are a priori assumed, it is easy to find stationary valued (Y, Z ), and we come to the preceding oscillating solution. At the same time, near thermodynamic equilibrium (i.e., at R aa P), there exits only a sole and stable stationary state of the system with (Y Z R). 

The eigenvalues a represent the possible measured values of the variable A. The Schrodinger equation (2.38) is the best-known instance of an eigenvalue equation, with its eigenvalues corresponding to the allowed energy levels of the quantum system. 

Equation (1.116) is an eigenvalue equation, and andE are eigenfunctions and corresponding eigenvalues of the Hamiltonian. If at time t = Q the system is in a state which is one of these eigenfunctions, that is,... 

For such a coupled spin system the density matrix is conveniently expressed in the product space of the eigenfunctions of the 1 operators of the uncoupled spins. Denoting the eigenfunctions for the spin-up and the spin-down states as a) and 1/6), respectively, the following eigenvalue equations apply for each of the coupled spins. 

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