Double spacing

Figure 2(b) represents the potential surface of the identical system, mapped onto the double-cover space [28], The latter is obtained simply by unwinding the encirclement angle < ), from 0 2ti to 0 4ti, such that two (internal) rotations around the Cl are represented as one in the page. The potential is therefore symmetric under the operation Rin defined as an internal rotation by 2n in the double space. To map back onto the single space, one cuts out a 271-wide sector from the double space. This is taken to be the 0 2ti sector in Fig. 2(b), but any 27i-wide sector would be acceptable. Which particular sector has been taken is represented by a cut line in the single space, so in Fig. 2(b) the cut line passes between < ) = 0 and 2n. Since the single space is the physical space, any observable obtained from the total (electronic + nuclear) wave function in this space must be independent of the position of the cut line. 

Figure 3. (a) The unsymmetrised nuclear wave function (soUd line) in the double space. The... 

Figure 4. (a) Single- and (b) double-space representations of 4 e (solid) and bo (dashed) for a system that does not encircle the Cl. 

So far, we have treated the atoms as distinguishable particles, both in the general theory of Section II and in the application to H + H2 in Section III. Here, we explain how to incorporate the effects of particle exchange symmetry. First, we discuss how the symmetry of the system maps from the physical onto the double space, and then explain what effect the GP has on wave functions of reactions that (like H + H2) have identical reagents and products. 

A useful property of the double space is that it clarifies the treatment of symmetry [28]. In the single space, the symmetry of Fq can appear confusing, because it depends on the position of the cut line. One way to avoid this confusion is to consider the symmetry of the total (electronic + nuclear) wave function L P, which is of course independent of the position of the cut line [6, 7]. Another way is to map Fg onto the double space. 

In Section II, we explained that and Fq are respectively symmetric and antisymmetric under the operator. R271 in the double space. More generally, if the... 

Figure 16, Relation between symmetry in (a) the single space and (b) the double space of a system whose molecular symmetry group has a a plane in the single space, and is isomorphic with C2v in the double space.

We can represent this function in the single space, provided we use a common cut line for all three components. This is shown schematically in Fig. 17. Use of the common cut line is equivalent to taking the linear combinations in the double space, then cutting a 27i-wide section out of the entire (4)). The winding numbers n of the Feynman paths that enter the three equivalent reagent channels must all be dehned with respect to the common cut line, since they are analogous to paths starting at different points in the initial state of a unimolecular reaction (Section 11.D). 

Figure 18. Complete unwinding of an encircling nuclear wave function >0 by mapping onto higher cover spaces, (a) The function in the single space (b) e in the double space (c) 4 in the quadruple space (d) schematic picture of in a 2hn cover space. In each case, will be completely unwound if it contains contributions from Feynman paths belonging to (b) 2, (c) 4, and (d) h different winding-number classes.

When higher n Feynman paths contribute to the wave function, one has simply to apply repeatedly the single- to double-space mapping, until the nuclear wave function is completely unwound (in the sense just defined). Thus, if the wave function contains only n = —2, —1,0,1 paths, then we need to compute a function in the double space that satisfies the boundary condition 4 (([)) = —(cj) + 4ti). Adding this function to the [which satisfies he(( )) = he(( ) + 4ti)] then gives a new function, [ 4(( )), which occupies the quadruple space (j) = 0 8ti (see Fig. 18c). This new quadruple-space wave function will be completely unwound, such that there is a gap between its clockwise and counterclockwise branches. The n= 2, 1,0,1 contributions will lie in the... 

To clarify, the complete unwinding of the wave function is not required to explain the effect of the GP. The latter affects only the sign of the odd n Feynman paths with respect to the even n paths, and is thus explained completely once one has unwound these two classes of path by mapping onto the double space. The complete unwinding explains the interference within the even n and odd n contributions, by unwinding each of them further, into the contributions from individual values of n. 

To see why this is so, let us attempt to apply the procedure of Section II.B to a bound-state wave function. This is illustrated schematically in Fig. 19. It is clear immediately that we cannot construct an unsymmetric in the double space, because each bound-state eigenfunction must be an irreducible representation of the double-space symmetry group. Thus a bound-state function in the double space is necessarily symmetric or antisymmetric under R2k, and is thus either a Fq or a Fn function. For a Fq function, we have Fn = 0 (since and Fn cannot form a degenerate pair), which implies [from Eq. (6)] that... 

Figure 19. Relation between and for a bound-state system. The functions in the single space (a) can be mapped onto the double space (b) where they have opposite symmetries under R2n, and belong to different symmetry blocks of the double-space Hamiltonian matrix (c). Unhke reactive wave functions, bound-state functions cannot be unwound from around the Cl.

Of course, the distinction between reactive- and bound-state wave functions becomes blurred when one considers very long-lived reactive resonances, of the sort considered in Section IV.B, which contain Feynman paths that loop many times around the CL Such a resonance, which will have a very narrow energy width, will behave almost like a bound-state wave function when mapped onto the double space, since e will be almost equal to Fo - The effect of the GP boundary condition would be therefore simply to shift the energies and permitted nodal structures of the resonances, as in a bound-state function. For short-lived resonances, however, Te and To will differ, since they will describe the different decay dynamics produced by the even and odd n Feynman paths separating them will therefore reveal how this dynamics is changed by the GP. The same is true for resonances which are long lived, but which are trapped in a region of space that does not encircle the Cl, so that the decay dynamics involves just a few Feynman loops around the CL... 

The formalism can be extended for a quantum Jield with the TFD Lagrangian density given by t = — , where is a replica of for the tilde fields so leading to similar equations of motion. For the purpose of our applications, we shall restrict our analysis to free massless fields. Thus, considering the free-massless boson (Klein-Gordon) field, the two-point Green function in the doubled space is given by... 

To introduce temperature we use the thermofield dynamics (TFD) formalism (Takahashi et.al., 1996 Das, 1997). TFD is a real time finite-temperature field theory. In TFD the central idea is the doubling of the Hilbert space of states. The operators on this doubled space... 

Only equation (25), which is the counterpart of equation (21) in the subspace of Asd,sd is solved iteratively. In this equation all the triples components have been projected into the singles-and-doubles space. Once equation (25) has been solved the C in) solution vector is stored on disk. The triples component of the solution vector, Cj(n), may then be constructed on the fly (see equation (26)), whenever it is needed. In this way, the storage of the triples components is avoided. 

How does this relationship affect your writing For most situations, it is in your best interest to be formal (but not stuffy), respectful (but not overly gracious), and courteous (but not ceremonious). You must also follow the provided guidelines or expectations. For example, if your instructor wants your essay typed in a 12-point font, double-spaced, with one-inch margins, and one staple in the top left-hand corner, that s exactly what you should hand in. 

References in submitted manuscripts are double-spaced (like the rest of the text), even though the references will be single-spaced in the final publication. 

Spacing Double spacing required throughout most of the manuscript... 

Quickplan and perform directed creation for a two- to four-page, double-spaced, handwritten essay on the topic What Are My Aspirations and Goals for the Next Five Years ... 

Elliot, R. J. (1954) Spin-orbit coupling in band theory - character tables for some double space groups. Phys. Rev. 96, 280-7. 

Hurley, A. C. (1966) Ray representations of point groups and the irreducible representations of space groups and double space groups. Phil. Trans. Roy. Soc. (London) A260, 1-36. 

It is convenient to introduce a Liouville space, or double space, that is a direct product of cap and tilde spaces. In Liouville space, operators are considered to be vectors and Hilbert-space commutators are considered to be operators. Equation (48) is then expressed as... 

Double-spaced, face-centered lattice of chemisorbed oxygen in (110) azimuth at 17 volts. Multiply current scale by 6... 

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