Displacement operator

Each of the factorized operators are displacement operators and can thus be applied seriatim to the initial state vector to give the final solution. 

Thus, a 2-D separation can be seen as 1-D displacement operating in two dimensions. The 2-D TLC separation is of no interest if selection of the two mobile phases is not appropriate. With this in mind, displacement in either direction can be either selective or non-selective. A combination of two selective displacements in 2-D TLC will lead to the application of different separating mechanisms in each direction. As an extreme, if the solvent combinations are the same (5ti = 5t2 5vi = va) or very similar (5ti 5vi 5ya), the compounds to be separated will be poorly resolved or even unresolved, and as a result a diagonal pattern will be obtained. In such circumstances, a slight increase in resolution might occur, because of an increase by a factor of V2 in the distance of migration of the zone (4). 

If D( ) is a linear displacement operator that converts a function [Pg.392]

Here we have uncovered an interesting connection between the linear displacement operator D and the linear momentum operator P ... 

Let < j(k) be the Klein-Gordon amplitude corresponding to a spin zero particle localized at the origin at time t = 0. Since in momentum space the space displacement operator is multiplication by exp (— tk a), the state localized at y at time t = 0 is given by exp (—ik-y) (k). This displaced state by condition (b) above must be orthogonal to (k), i.e. 

For n 0 the above equation must coincide with the equation of motion, hence P0 is the hamiltonian of the system. Similarly one deduoes that the P, (i = 1,2,3) must be the components of the total momentum operator. We call the operator PB the displacement operators for spaoe time translations in the sense that for an arbitrary operator F(x) which is a function of tfi(x) and A (x)... 

Downward displacement operator, 399 Drell, S. D., 724 Druyvesteyn distribution, 49 Druyvesteyn, M. J49 Dual problem in linear programming, 304... 

Linear algebraic problem, 53 Linear displacement operator, 392 Linear manifolds in Hilbert space, 429 Linear momentum operator, 392 Linear operators in Hilbert space, 431 Linear programming, 252,261 diet problem, 294 dual problem, 304 evaluation of methods, 302 in matrix notation, simplex method, 292... 

Upward displacement operator, 398 Vrdbe, M, 363 Utility matrix, 314... 

In the Heisenberg representation a time-dependent dipole operator p(t) is generated from its value at some previous time t by a unitary transformation with the time-displacement operator exp — t )/h, so that... 

Here Hint is the anharmonic interaction in the collinear-configurational approximation (see Refs. [5,7]), Vmk = v/n,7i/2atoms with respect to the atom(s)of the mode (Y.i eimeim = M- We take into account that the strongly excited mode can be considered classically, and replace its coordinate operator by Q(t) = A cos( )/f), where A is the initial amplitude of the mode. Then... 

If the anharmonic interaction is not weak, then D(t) should be found from the equation of motion, which in case k > 2 turns out to be a non-linear integral equation. To get the corresponding equation, one should start from the equation of motion for the displacement operator(s) q. From equation (5) it follows that... 

If we assume the diatom to be harmonic, the matrix element of the displacement operator in (12.9) can be shown to yield (Cohen-Tannoudji, Diu, and Laloe 1977 ch.V)... 

In Sections III—VI the nuclear displacement operators H n) are treated as perturbation operators and response function theory is used to determine the implicit geometry dependence of the wave function to each order in the nuclear displacement ... 

When one phase is displaced by another in a porous medium, instabilities may develop that allow the displacing phase to finger through the displaced phase, bypassing major portions of it. For this reason, sweep control measures are very important to the success of commercial displacement operations. 

Figure 6. Schematics of the three common regenerative cryocoolers. The Stirling eryocooler (a) uses a valveless compressor or pressure oscillator and has a moving displacer operating synchronously with the piston. The pulse tube eryocooler (b) has no displacer in the cold head. The Gifford-McMahon eryocooler (c) uses a valved compressor with oil lubrication and oil removal equipment.

The total strain-displacement operators B and B of all nodes are assembled by B and B. Subsequently the complete strain-displacement operator B = [Bq Bi] is partitioned corresponding to the interfaces j = 0 and j = I into the matrices Bq and Bi ... 

Following elution of the isotachic train and the displacer solution from the column, the column must be regenerated and reequilibrated with the carrier before any subsequent displacement separation. This reequilibration step can be lengthy and is frequently considered a major Umitation to efficient displacement operation. Displacement chromatography requires the competitive isotherms of the solutes and the displacer to be convex upward and to not intersect each other. (See the entry Distribution Coefficient for related information.)... 

When the solute isotherms cross one another, the situation becomes more complex. It then becomes possible to experience selectivity reversal that is, at one displacer concentration, the solutes elute in the order A first, then B, whereas at another displacer concentration, the order is B first, then A. In a study of this problem, Antia and Horvath showed the existence of the separation gap. This is a region in the isotherm plane, the position of which depends on the ratio of the saturation capacities of the solutes in question. If the operating line is outside the separation gap, displacement occurs in the normal fashion. The elution order of the solutes then depends on the position of the operating line relative to the separation gap. However, if the operating line is within the separation gap, displacement operation does not separate the displaced solutes, but results in the elution of a mixture of the solutes. 

Kanamori in his seminal work [3] suggested a more general approach called the displacement operator method. According to this method the initial bk phonon operators are replaced in the Hamiltonian (1) with the ak new ones like that... 

Now, we see that if we replace the operator B by the Hamiltonian H of a simple oscillator, these equations are identical to the corresponding equations of the simple quantum oscillator [30, 31]. According to this strong analogy, we are able to determine the amplitude matrix and the matrices of the Bohlin operator B, the displacement operator and the momentum operator. The results are as follows ... 

Resulting from Equation (52) using Equation (56), the matrix of the displacement operator in energy representation has the form... 

Here, the displacement operator was used in a narrow sense. Next, we are going to solve the... 

Displacement operator coherent states are those states which are created from the ground state by a particular unitary displacement operator. 

Glauber [8] constructed coherent states in the ID Hilbert space by applying the displacement operator D a, a ) = exp(afit — a a) on vacuum state 0). Analogously, one can define the generalized coherent state [16]... 

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