Fourier representation

In this Fourier representation the Hamiltonian is quadratic and the equipartition theorem yields for the thennal... 

Let us eonsider the Fourier representation of f(t) in terms of the eomplex exponentials introdueed above. For an arbitrary periodie funetion f(t), we ean write... 

In order to find the effect of broadening of the surface on the structure parameters H and K, we first study the ordered phases with the diffusive interfaces. The ordered phases can be described by the periodic surfaces (0(r)) = 0 and we can compare and with H and K. The numerators in the definitions (76) and (77) in the Fourier representation assume the forms [68]... 

Here, /j f and Xi,f are the initial and final state electronic and vibration-rotation state wavefunctions, respectively, and i,f are the respective state enei gies which are connected via a photon of energy boo. For a particular electronic transition (i.e., a specific choice for /i and /f and for a specific choice of initial vibration-rotation state, it is possible to obtain an expression for the total rate Rj of transitions fi om this particular initial state into all vibration-rotation states of the final electronic state. This is done by first using the Fourier representation of the Dirac 5 function ... 

The Fourier representation of the Dirac delta function leads then to the result... 

For very stiff pulling springs (where r is almost a coupling parameter under external control), we can instead pursue the so-called stiff-spring approximation of Park et al. [45]. A Fourier representation of the spring Boltzmann factor on the right-hand side of (5.60) results in... 

If we are interested in the off-diagonal elements of the density matrix, that is p(x, x j3), the paths are no longer cyclic but begin at x and terminate at x1. Then the Fourier representation of the path is... 

Greek indices a, p = x,y,z of the Cartesian coordinate axes is meant). Minimization of expression (2.1.1) for an arbitrary two-dimensional Bravais lattice becomes possible since the Fourier representation in q implies the reduction of the double sum over j and/ to the single sum over q. Then the ground state energy is given by... 

When we use the Fourier representation of the Oseen tensor G, Ay is given by... 

The research was greatly facilitated by two important elements. The (formal, perturbative) solution of the Liouville equation is greatly simplified by a Fourier representation (see Appendix). The latter allows one to easily identify the various types of statistical correlations between the particles. The traditional dynamics thus becomes a dynamics of correlations. The latter is completed by... 

The Fourier representation provides us with a particular realization of this decomposition The vacuum Vp is realized by the Fourier component Po,o,...,o(7) of the distribution function. The abstract decomposition (4) can, however, also be realized in more general ways. Equation (7) expresses the fact that in a system without interactions, one cannot create correlations out of the vacuum. 

In Section XI-A we described the mixture of zero-order states that can be thought of as the generalized Fourier representation of a true eigenstate of a molecule in which a discrete zero-order state is coupled to... 

Order-parameter fluctuations can be generalized by introducing the wave-vector f3 in a Fourier representation,... 

Fig. 7.81. Comparison of output of on-off element with those produced by first and second terms of corresponding Fourier representation...

Fourier representation of electron density suggests the possibility of direct structure analysis. If all structure factors, F(hkl), are known, p(xyz) can be computed at a large number of points in the unit cell and local maxima in the electron-density function are interpreted to occur at the atomic sites. A typical single-crystal diffraction pattern of the type used for measuring structure factor amplitudes is shown in Figure 6.12. 

The general solution of Eq. (18) is very complex, but for slab symmetry, if the heat effect of micropore formation is very short (that allows the application of perturbation method), the solution has the Fourier representation form [5]. 

In these the functions and their derivatives are evaluated at the points (0, y, 0). Solution of the (still) partial differential equations, Eq. (92), is promoted by using Fourier representations for the functions, il/ n involving the wavelength, 2n/k... 

A general remark is in order concerning the adoption of C q) in the framework of the present Fourier representation. The characteristic ratio is usually evaluated on a nonperiodic chain, in contrast with the Fourier representation. In the case of short chains this may entail minor errors deriving from end effects, which vanish if N is large enough that <1(1).1(N)>o F [22]. 

Fourier and Floquet space. The definitions of the different block-elements are given in a for the Fourier representation and in b for the Floquet representation. The Hamiltonians retain the same general form, but the time-dependent elements in a become time-independent in b with the addition of the number operator elements on the diagonal... 

A schematic of the Fourier representation is given in Fig. 14. The fact that the Hamiltonian is still time dependent does not show any advantage of this representation. However, things can be simplified when we use the commutation relations between the number and ladder operators ... 

The Fourier representation of the anharmonic propagator is obtained from Eqs. (73) and (72), by using Wick s theorem (69), together with the Fourier representation of the harmonic phonon propagators. The time integrations can then explicitly be performed, yielding the condition 2,- w, = 0 and a factor /3 at every interaction vertex. 

Each term in this expansion has a Fourier representation of the form... 

---