Invariance with respect to rotation

The Hamiltonian is also invariant with respect to any rotation in space U of the isotropy of coordinate stem about a fixed axis. The rotation is carried out by apptying an or-thogonal matrix transformation V of vector r = (x, y, z) that describes any particle of coordinates x, y, z. Therefore all the particles undergo the same rotation and the new coordinates are r = Ur = Ur. Again there is no problem with the potential energy, because a rotation does not change the interparticle distances. 

What about the Lapladans in the kinetic energy operators Let us see. 

one has invariance of the Hamiltonian with respect to any rotation about the origin of the coordinate system. This means (see p. 955) that the Hamiltonian and the operator of the square of the total angular momentum (as well as of one of its components, denoted by J,) commute. One is able, therefore, to measure simultaneously the energy, the square of total angular momentum as well as one of the components of total angular momentum, and (as it will be shown in (4.6)) one has 

Any rotation matrix may be shown as a product of elementary rotations, each about axes x, y or z. For example, rotation about the y axis by angle 0 corresponds to the matrix 

The pattern of such matrices is simple one has to put in some places sines, cosines, zeros and ones with the proper signs. This matrix is orthogonal, i.e. = U, which you may easily check. The product of two orthogonal matrices represents an orthogonal matrix, therefore any rotation corresponds to an orthogonal matrix. 

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The drawback of this approach is that it makes the integrals non-invariant with respect to rotations of the coordinate frame. The source of invariance is that the fictitious charge configurations have non-vanishing higher multipole momenta due to... 

This means that scalar operators are invariant with respect to rotations in coordinate or spin space. An example for a scalar operator is the Elamiltonian, i.e., the operator of the energy. 

Here the coefficients a and b characterize the surface-energy anisotropy and can be computed from the surface-energy dependence on the surface orientation. Naturally, the nonhnear operator Too i is invariant with respect to rotations by 7t/2, as well as any of the transformations x —s- —x, y — —y, x y, while Finis invariant with respect to rotations by 27t/3 as well as the transformation y —s- —y, b —b. The functions Wo 2,zih) are determined by the type of a wetting interaction model and can also differ for different orientations of the film surface. 

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

The 4D bispectrum is invariant with respect to rotations of four-dimensional space, which include three-dimensional rotations. However, there are additional rotations, associated with the third polar angle Qq, which, in our case, represents the radial information. In order to eliminate the invariance with respect to the third polar angle, we modified the atomic density as follows ... 

A material is mechanically isotropic if all of its mechanical properties are the same in all spatial directions. The elasticity tensor must thus remain unchanged by arbitrary rotations of the material or the coordinate system. Its components must be invariant with respect to rotations. 

A very important aspect of setting up a neural network potential is the choice of the input coordinates. Cartesian coordinates cannot be used as input for NNs at all, because they are not invariant with respect to rotation and translation of the system. Since the NN output depends on the absolute numbers fed into the NN in the input nodes, simply translating or rotating a molecule would change its energy. Instead, some form of internal coordinates like interatomic distances and bond angles or functions of these coordinates should be used. To define a non-periodic structure containing N atoms uniquely, 3N-6 coordinates are required. However, for NNs redundant information does not pose a problem, and sometimes the complete set of N(N— l)/2 interatomic distances is used." ... 

To proceed further, we shall consider a simple system for which results can be obtained in a closed form. Let medium 1 be vacuum (cj = 1) and medium 2 be a jellium. The jellium is invariant with respect to rotations around the surface normal and hence its dielectric tensor has the nonzero components Cxx = Cyy = C and Czz- Also, the translational invariance along the surface ensures that e(r, r ) depends on the difference between the lateral coordinates, i.e.. 

The bifurcations described above are subject to the condition of invariance with respect to rotation around the z-axis. The straight-line r = 0 is then an integral curve, and in the case where Oi and O2 are both saddles, this is... 

As the rate of rotation increases, the rotational spinning sidebands move further out and become weaker. At very high spinning speeds, the intensities of the sidebands become negligible, and the spectrum consists of the narrowed central line at the Larmor frequency, coq. The intensities of the first satellites are expected to decrease at a rate of a>, thus preserving their contributions to the second moment of the entire spectrum. Indeed, the magnirnde of the second moment of the spectrum should be invariant with respect to rotation. The intensities of the second and higher satellites fall even more rapidly with an increase in cur, as cu " for the nth satellite. 

Despite the derivation of simple analytical formulas, the above analysis shows that the orientation-averaged extinction and scattering cross-sections for macroscopically isotropic and mirror-symmetric media do not depend on the polarization state of the incident wave. The orientation-averaged extinction and scattering cross-sections are invariant with respect to rotations and translations of the coordinate system and using these properties, Mishchenko et al. [169] have derived several invariants of the transition matrix. 

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