Invariance with respect to translation

Transformation Z = r+T does not change the Hamiltonian. This is evident for the potential energy V, because the translations T cancel, leaving the interparticle distances unchanged. For the kinetic energ one obtains 

The Hamiltonian is therefore invariant with respect to any translation of the coordinate stem. 

No matter, whether the coordinate system used is fixed in Trafalgar Square, or in the centre of mass of the system, one has to solve the same mathematical problem. 

The solution to the Schrodinger equation corresponding to the space fixed coordinate system (SFS) located in Trafalgar Square is pN, whereas is calculated in the body-fixed coordinate system (see Appendix I) located in the centre of mass at Rcm with the (total) momentum PcM- These two solutions are related by i pN = cm)- The number A = 0,1,2. counts the 

This means that the energy spectrum represents a continuum, because the centre of mass may have any (non-negative) kinetic energy p jj /(2m). If, however, one assumes that pcM = const, then the energy spectrum is discrete for low-energy eigenvalues (see eq. (1.13)). 

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A range of physicochemical properties such as partial atomic charges [9] or measures of the polarizabihty [10] can be calculated, for example with the program package PETRA [11]. The topological autocorrelation vector is invariant with respect to translation, rotation, and the conformer of the molecule considered. An alignment of molecules is not necessary for the calculation of their autocorrelation vectors. 

For a free noninteracting spinning particle, invariance with respect to translations and rotations in three dimensional space, i.e., invariance under the inhomogeneous euclidean group, requires that the momenta pl and the total angular momenta J1 obey the following commutation rules... 

The topological autocorrelation vector is invariant with respect to translation, rotation and the confotmer of the molecule considered. An alignment of molecules is not necessary for the calculation of their autocorrelation vectors. 

We remark that we wrote the same Hi in (3.109c) and in (3.109d) so that wi does not contain the mode u. This ought to be the case, because in the moving coordinate system (3.95), the system is no longer invariant with respect to translations in x, and therefore the mode vf which appears due to translation invariance should be absent. 

The Non-Relativistic Hamiltonian and Conservation Laws Invariance with Respect to Translation Invariance with Respect to Rotation... 

The one-electron Hamiltonian is invariant with respect to translations by any lattice vector. Therefore, its... 

The Hamiltonian is invariant with respect to translations by any lattice vector. Therefore its eigenfunctions are simultaneously eigenfunctions of the translation operators (Bloch theorem) i((r — Rj) = esp(—ikRj)4>ic(r) and transform according to the irreducible representation of the translation group labelled by the wave vector k. 

The unknown amplitudes a and 3 are determined by solving a system of ordinary differential equations, called the bifurcation equations. In [4,5], it is shown that the symmetries present in the governing equations (2.1) and the eigenfunctions of equations (2.3) are reflected in corresponding symmetry properties of the bifurcation equations. Specifically, since the problem is invariant with respect to translations in T and Y, and with respect to reflections in Y, the analysis leads to bifurcation equations of the form... 

It follows from the definition of the functionals and Ti that the exchange-correlation functional is invariant under the rotation and translation of the electron density. This follows directly from the fact that the kinetic energy operator f and the two-electron operator W are invariant with respect to these transformations. This also has implications for the exchange-correlation functional. 

Fig. 5.5. Geometrical structure of a close-packed metal surface. Left, the second-layer atoms (circles) and third-layer atoms (small dots) have little influence on the surface charge density, which is dominated by the top-layer atoms (large dots). The top layer exhibits sixfold symmetry, which is invariant with respect to the plane group p6mm (that is, point group Q, together with the translational symmetry.). Right, the corresponding surface Brillouin zone. The lowest nontrivial Fourier components of the LDOS arise from Bloch functions near the T and K points. (The symbols for plane groups are explained in Appendix E.)...

Nevertheless, there exists finite-additive probability which is invariant with respect to Euclidean group E(n) (generated by rotations and translations). Its values are densities of sets. 

If the interaction between two objects does not depend explicitly on the lime coordinate, then the actions lhal take place do nol depend on when one starts to measure lime i.e.. the properties of the system are invariant with respect to a translation of the origin of coordinates along the lime axis. This symmetry is associated with conservation of energy. Use of a 4-dimen.sional coordinate system allows one to associate conservation of momentum and energy in a unified manner with the geometrical symmetry of space-time. 

The relative nuclear configuration RNC Xk( ), Zk, Mk) is defined as the set of informations determining a NC up to translations and rotations in 3 3, i.e. invariant with respect to transformations of the inhomogeneous three-dimensional rotation group 10(3). Conveniently the RNC is determined by internal structural parameters, 2.13K-6 which are invariant with respect to (w.r.t.) 10(3). 

Till now, we have only considered a mathematical set of points. However, a material, in reality, is not merely an array of points, but the group of points is a lattice. A real crystalline material is constituted of atoms periodically arranged in the structure, where the condition of periodicity implies a translational invariance with respect to a translation operation, and where a lattice translation operation, T, is defined as a vector connecting two lattice points, given by Equation 1.1 as... 

While the elements of the tensor P(s), Eqs. 5 and 6, are numerically invariant with respect to a parallel translation of the reference axis system, they do depend on the orientation (the directions) of the reference axes. If the mass point system is rigidly rotated by an orthogonal transformation T (with Tt = T-1),... 

According to Noether s theorem (Arnold (1989)) symmetries of a mechanical system are always accompanied by constants of the motion. According to Section 3.1, system symmetries can be obvious (e.g. geometric) or hidden . Examples for obvious symmetries that lead to constants of the motion are invariance with respect to time translations, spatial translations and rotations. Invariance with respect to time leads to the conservation of energy, spatial and rotational symmetries lead to the conservation of linear and angular momentum, respectively (see, e.g., Landau and Lifschitz (1970)). Hidden symmetries cannot be associated with... 

When two points in the same Cartesian basis are related to one another by a translation, then the coordinates of the second point are invariant with respect to a different Cartesian basis, in which the orientations of the axes remain the same as in the first basis but its origin is shifted along the three non-coplanar vectors, tx, ty and t, as shown in Figure 1.49. 

Due to the relative character of the number operator — /S/S0, all the physical predictions of the Floquent model must be invariant with respect to a global translation of the relative photon numbers. We show that this is indeed the case for the properties discussed above. The propability P(L, t) is independent of the particular initial photon number state chosen that is, it is independent of k since... 

While the system is invariant to any translation of the kink along the chain in the continuum limit (hence the kink energy Ek resembles the energy of a free particle), discrete models only need to be invariant with respect to a translation by a lattice constant b. If the chain is commensurate with the substrate and/or k is sufficiently small, then a suitably defined kink coordinate X [109] will experience a potential periodic in / thus... 

A fundamental characteristic of spatially periodic systems is the existence of a group of translational symmetry operations, by means of which the repeating pattern may be brought into self-coincidence. The translational symmetry of the array, expressing its invariance with respect to parallel displacements in different directions is represented by a lattice. This lattice consists of an array of evenly spaced points (Fig. 3-13), such that the structural elements appear the same and in the same orientation when viewed from each and every one of the lattice points. Another important property of spatially periodic arrays is the existence of two characteristic length scales, corresponding to the average microscopic distance between lattice... 

In what follows we will use the cyclic boundary conditions for the wave-functions. By this condition a wavefunction F remains invariant with respect to a translation by any of the integer-valued vectors N ai, AL.a ., where... 

From the invariance of the action S with respect to translations in 4-coordinate space follow the expressions for the energy and momentum densities of the electron-positron field ... 

Having thus argued in support of the idea of self-similarity with respect to scaling, we now draw attention to a separate class of problems to which a different type of self-similarity is appropriate, namely, self-similarity with respect to translation of the generalized time. A function f t fo), subject to the initial condition /g = /(O /g), is said to be translationally invariant if it satisfies the functional relationship... 

Thus, it turns out that invariance of the equation of motion with respect to an arbitrary translation in time (time homogeneity) results in the energy conservation principle with respect to translation in space (space homogeneity) gives the total momentum conservation principle and with respect to rotation in space (space isotropy) implies the total angular momentum conservation principle. 

If the Hamiltonian turned out to be invariant with respect to a symmetry operation U (translation, rotation, etc.), this would imply the commutation of U and H. We will check this in... 

It is easy to see that the operators T(Ri) form a group (see Appendix C available at booksite. elsevier.com/978-0-444-59436-5, p. el7) with respect to their multiplication as the group operation. 43 jjj Chapter 2, it was shown that the Hamiltonian is invariant with respect to any translation of a molecule. For infinite systems, the proof looks the same for the kinetic energy operator, file invariance of V is guaranteed by Eq. (9.2). Therefore, the effective one-electron Hamiltonian commutes with any translation operator ... 

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