Eigenstates, normalization

The probability of observing a partieular value fk when F is measured, given that the system wavefunetion is P prior to the measurement, is given by expanding P in terms of the eomplete set of normalized eigenstates of F... 

Thus, the expansion of / in terms of eigenstates of the property being measured dietated by the fifth postulate above is already aeeomplished. The only two terms in this expansion eorrespond to momenta along the y-axis of 2h/Ly and -2h/Ly the probabilities of observing these two momenta are given by the squares of the expansion eoeffieients of / in terms of the normalized eigenfunetions of -ihd/dy. The funetions (l/Ly)F2 exp(i27iy/Ly) and... 

This contribution considers systems which can be described with just the Hamiltonian, and do not need a dissipative term so that TZd = 0- This would be the case for an isolated system, or in phenomena where the dissipation effects can be represented by an additional operator to form a new effective non-Hermitian Hamiltonian. These will be called here Hamiltonian systems. For isolated systems with a Hermitian Hamiltonian, the normalization is constant over time and the density operator may be constructed in a simpler way. In effect, the initial operator may be expanded in its orthonormal eigenstates (density amplitudes) and eigenvalues Wn (positive populations), where n labels the states, in the form... 

The value of P is important only with respect to technical considerations. The presence of finite orbital overlap between the initial and final states shghtly alters the p dependence of the relationship between Ft and A, The overlap corrections for the instanton analysis arise from the slightly different normalization of the symmetric and antisymmetric eigenstates, which can be obtained from the simple model in Fig. 2. The coefficients of the symmetric and antisymmetric eigenstates in terms of the zeroth order states, nominally + become instead... 

The 2-RDM is automatically antisymmetric, but it may require an adjustment of the trace to correct the normalization. The functionals in Table I from cumulant theory allow us to approximate the 3- and the 4-RDMs from the 2-RDM and, hence, to iterate with the contracted power method. Because of the approximate reconstruction the contracted power method does not yield energies that are strictly above the exact energy. As in the full power method the updated 2-RDM in Eq. (116) moves toward the eigenstate whose eigenvalue has the largest magnitude. 

Let the eigenvalue w be fixed and assume that fit is nondegenerate and unit-normalized. The restriction to nondegenerate eigenstates will be relaxed in Section V, but for now we consider only pure-state density matrices. The A -electron density matrix for the pure state fit is... 

The wave function of the ion that remains after annihilation is a superposition of eigenstates of the Hamiltonian of the ion, the relative probabilities of which may be determined from the wave function used in the calculation of Zeg. The annihilation process takes place so rapidly, compared with normal atomic processes, that it is reasonable to assume the validity of the sudden approximation. Consequently, the wave function of the residual ion when the positron has annihilated with electron 2 at the position r = r2 is... 

Resonances unassociated with eigenstates of Feshbach s QHQ are often associated with the shape of some effective potential in an open channel, normally a combination of short-range attractive and long-range repulsive potentials, forming a barrier, within which a large part of the wavefunction is kept. These resonances are called "shape resonances" or "potential resonances." They occur at energies above and usually close to the threshold of that open channel. 

The eigenstates associated with the asymptotic free relative motion, the so-called plane waves, (R p) = (2nh) 3/2exp( p R/h) (with delta-function normalization on the momentum scale), can be expanded in terms of the common eigenstates of L2, and Lz, i.e.,... 

For a single excitation path (i.e., one or three photons), the only possibility for controlling the outcome of the reaction is to select the excited eigenstate by varying E, as is normally done in mode-selective processes. A completely new form of control becomes possible, however, if both excitation paths are simultaneously available. In that case, the reaction probability is... 

The coefficients Ck are real and Skk1 =< k k > is the overlap of normalized VB diagrams with identical electron distributions rij. Normalization illustrates the general problem of finding matrix elements between correlated states. We express an operator in second-quantized notation and consider exact eigenstates i> > and x > that may be in the same or different symmetry subspaces. The matrix elements Akk of A are obtained as shown in (11) to give... 

The V-B coupling Hamiltonian to first order in the three HOD dimensionless normal coordinates is Hv b = —2, c], l , where F, is the inter-molecular force due to the solvent exerted on the harmonic normal coordinate, evaluated at the equilibrium position of the latter. This force obviously depends on the relative separations of all molecules, and on their relative orientations. In the most rigorous quantum description of rotations, this term would depend on the excited molecule rotational eigenstates and of the solvent molecules. Instead rotation was treated classically, a reasonable approximation for water at room temperature. With this form for the coupling, the formal conversion of the Golden Rule formula into a rate expression follows along the lines developed by Oxtoby (2,53), with a slight variation to maintain the explicit time dependence of the vibrational coordinates (57),... 

Each molecular vibration factor in Equation (3) is a type of molecular time correlation function for the internal vibrational dynamics. In the harmonic approximation, i) and f) would reduce to the harmonic vibrational eigenstates and the qj would be the actual molecular normal modes. Then one has the simplification... 

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