Differentiability expectation value

The ordinary differential equation (270) can now be solved by employing the boundary conditions (273)-(276) to generate the expected value of... 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

Muda and Hanawa (MH) have approached the problem by considering the time variation of the quantities (r) = < r) cl,c, t)>, which are expectation values of products of creation and annihilation operators for site-centered orbitals >. The Schrodinger equation then leads to a set of first-order differential equations, viz.,... 

The evaluation of the expectation values of other than r distances and their squares can be done by differentiating expressions (83) and (67), respectively. We will restrict ourselves to the case when m s are even, so that p = mu/2 with... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

Equation (9.32) is also useful to the extent it suggests die general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining die wave function, the Hellmann-Feynman theorem of quantum mechanics allows us to write... 

Deviation sum of squares is obtained by squaring the difference of expected values (Table 2.56) and experimentally obtained results (Table 2.54). By partial differentiation of ai bj and m and bringing it down to zero, we get the following system of normal equations ... 

M-clustering-induced collapse of short PEO chains because we know that polymer chains in bulk or in a very concentrated solution adopt a random coil conformation, as with the 0-temperature [70,71]. In addition, the solvent quality of water for PEO decreases as the solution temperature increases. In order to differentiate these two scenarios, the temperature dependence of (Rg)/(Rh) is plotted in Fig. 15 to reflect the chain density distribution. The fact that (Rg)/ R ) 1.1 at lower temperatures, instead of 1.5 (an expected value for linear coil chains), reflects its branching structure because short PEO chains have a length similar to the PNIPAM segments between two neighboring grafting points. The decrease of (Rg)/ Rh) from 1.0 to 0.5 clearly reveals a change of the chain conformation. 

Just as logarithms and exponentials are inverse operations, integration is the inverse of differentiation. The integral can be shown to be the area under the curve in the same sense that the derivative is the slope of the tangent to the curve. The most common applications of integrals in chemistry and physics are normalization (for example, adjusting a probability distribution so that the sum of all the probabilities is 1) and calculation of the expectation values of observable quantities. 

The differential cross-section refers, as in the classical description, to the scattering angle in the center-of-mass coordinate system. In order to relate to experimentally observed differential cross-sections, one has to transform to the appropriate scattering angle. This transformation takes the same form as discussed previously, essentially, because the expectation value of the center-of-mass velocity V is conserved just as in classical mechanics. 

Upon substituting the series expansion (9.48) into the previous equation, and equating the coefficients of 9i and 92, one has the following differential equations for the expected value functions ... 

So to obtain expectation values relevant to any particular experiment one needs an estimate of the density matrix at the time of measurement. For an NMR experiment, this typically requires the ability to estimate the time evolution of the density matrix for the pulse sequence used for the experiment. The time dependent differential equation that describes the time evolution of the density matrix, known as the Liouville-von Neumann equation is given by... 

The expected values of the yields are modeled by the differential equa-... 

Figure 9 Comparison of silicate mass fractions. Two assumptions for interior strueture are shown (i) differentiated—rock core, ice mantle, and (ii) homogeneous—uniformly mixed ice and roek. Also shown are silicate mass fractions for the Jupiter and Saturn systems and expected values for two models of the early solar nebula carbon chemistry (see text) (after Johnson et aL, 1987) (reproduced by permission of Ameriean Geophysieal Union from /. Geophys. Res. Space Phys. 1987, 92, 14884-14894).

In the course of differentiation of the Hamiltonian of Eq. (310) by components of R, all the electronic kinetic energy terms and all the terms describing electron-electron interactions are eliminated. Consequently, the force operator F is a one-electron operator and the expectation value can be written as... 

Note that the expectation value need not itself be a possible result of a single measurement (like the centroid of a donut, which is located in the hole ). When the operator 4 is a simple function, not containing differential operators or the like, then Eq (4.21) reduces to the classical formula for an average value ... 

Differentiating the expectation value (f 2(f) r) = (0 2 0) [see Eq. (40)] with respect to time and invoking the Schrodinger equation (atomic units)... 

This can be easily shown by noting that for a normalized wavefunction 4 the other terms arising from the differentiation of the energy expectation value vanish ... 

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