Translation operators

That is, the effect of a translational operation is detennined solely by the vector with components (kyj yj ) which defines the linear momenUim. 

Furthermore, since H, the hamiltonian, is the time translation operator... 

The special positions include the following points and those obtained from them by the translational operations of the face-centered lattice ... 

These are the four main operations required to define the symmetry of a crystal structure. The most important is that of translation since each of the other procedures, called symmetry operations, must be consistant with the translation operation in the crystal structure. Thus, the rotation operation must be through an angle of 2n / n, where n = 1, 2, 3, 4 or 6. 

From the above properties it is evident that the set of operations t forms a group, J the space group of the crystal. If the translation operations are the primitive translations is r , ... 

Equation (49) contains the Franck-Condon factors that are the matrix elements of the translation operator involved in the canonical transformation (36) with k = 1 that are given for m > n by... 

The entire cyclic voltammogram is no longer reversible according to the definition we have attached to this term so far. In other words, the symmetry and translation operations as in Figures 1.4 and 6.1 do no longer allow the superposition of the reverse and forward trace. It also appears that the midpoint between the anodic and cathodic peak potentials does not exactly coincide with the standard potential. The gap between the two potentials increases with the extent of the ohmic drop as illustrated in Figure 6.2 for typical conditions, which thus provides an estimate of the error that would result if the two potentials were regarded as equal. 

Several types of symmetry operations can be distinguished in a crystalline substance. Purely translational operations, such as the translations defining the crystal lattice, are represented by I 1, n3, with nu n2, n3 being integers. 

Proper rotational operations are represented by the n-fold rotation axes n 1000 (n = 2, 3,4, 6). Rotation-inversion axes such as the 2 axis are improper rotation operations, while screw axes and glide planes are combined rotation-translation operations. 

The unit 12-vector acts essentially as a normalized spacetime translation on the classical level. The concept of spacetime translation operator was introduced by Wigner, thus extending [100] the Lorentz group to the Poincare group. The PL vector is essential for a self-consistent description of particle spin. 

For a particle (including the photon) with mass, the spacetime translation operator P]> in the rest frame is... 

One-Dimensional Symmetries. These will not detain us long, but they do merit examination because they introduce in the simplest possible way several new concepts. There are seven types of ID symmetry, and they are shown in Figure 11.1, where a triangle is used as the motif. The simplest symmetry, class 1, is that having only the translation operation. The unit translation is the distance from any given point on one triangle to the identical point on the nearest triangle. 

Class 4 is obtained by introducing a transverse mirror line. Not only is this line reproduced by the translation operation, but a second set of transverse mirror lines is created. This is similar to what occurred in class 3. The second set is not equivalent to the first, but is brought into existence by it. 

Upon adding each one of these we create, respectively, the symmetries / 2, p3, p4, and p6. In each case it can be seen in Figure 11.7 that the combined effect of the explicitly introduced rotation axis and the translational operations generates further symmetry axes. Thus, inp2, in addition to the set of twofold axes explicitly introduced, two other sets, lying at the midpoints of the translations connecting them, arise automatically because of the repetitive nature of the lattice. In the case of p3 a second, independent set of threefold axes arises. The reader can see by inspection of Figure 11.7 the additional axes that arise in p4 and p6. 

Thus far we have addressed the symmetry of crystalline arrays only in terms of the proper rotations and the rotation-inversion operations (the latter including simple inversion, as 1, and reflection, as 2) that occur in point symmetries, along with the lattice translation operations. However, for a complete discussion of symmetry in crystalline solids, we require two more types of operation in which translation is combined with either reflection or rotation. These are, respectively, glide-reflections (or, as commonly called, glides) and screw-rotations. 

The action of the space-group operator (R v) c G (with R c P), the point group of the space group G on the Bloch function kfr) gives the transformed function vJk(r). To find the transformed wave vector k we need the eigenvalue exp ( ik -t) of the translation operator ( t). 

T and therefore equal to Jx, Jy, J without the common factor of 1/2 translation operator (the distinction between T a translation operator and T when used as a point symmetry operator will always be clear from the context)... 

This is very similar to creating and assigning variables, as addressed earlier in the word-problem section. In addition to identifying what is known and unknown, also take time to translate operation words into actual symbols. It is best when working with a word problem to represent every part of it, phrase by phrase, in mathematical language. 

In Section 4.2.2, we used the displacement (translation) operator exp(—ibpi/h). We consider here this operator and its action on the state ), i.e., we consider the momentum-space and coordinate-space representations of b) = exp(—ibpi/h) ). 

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