Invariance under transformation

The symmetry of a number of atomic quantities (wave functions, matrix elements, 3n./-coefficients etc.) with respect to certain substitution groups or simply substitutions like l — — / — 1, L — —L — 1, S — — S — 1, j — — j— 1, N 41 + 2 — N, v —y 41 + 4 — v leads to new expressions or helps to check already existing formulas or algebraic tables [322, 323]. Some expressions are invariant under such transformations. For example, Eq. (5.40) is invariant with respect to substitutions S — —S — 1 and v — 41 + 4 — v. Clebsch-Gordan coefficients in Table 7.2 are invariant under transformation j — —j — 1. However, applying this substitution to the coefficient in Table 7.2, we obtain the algebraic value of the other coefficient. 

Multi-Configurational Adiabatic Electron Transfer Theory and Its Invariance under Transformations of Charge Density Basis Functions. 

M. V. Basilevsky, G. E. Chudinov, and M. D. Newton, Chem. Phys., 179, 263 (1994). The Multiconfigurational Adiabatic Electron Transfer Theory and Its Invariance Under Transformations of Charge Density Basis Functions. 

Since the integral in Eq. (25.59) is a real physical quantity, its value cannot depend on the orientation of the coordinate system. Consequently, the integrand, 1/ /, must be invariant under transformations of the coordinate system. This invariance can obtain only if ij/ is invariant or merely changes sign under transformations of the coordinate system. 

Theorem 2 For a Coulomb Hamiltonian, in an actual configuration-interaction -type calculation with one-particle square-integrable, analytic functions as basis sets, all overlap, one- and two-particle integrals remain invariant under transformations of the type u r)... 

V is only invariant under transformations that interchange identical nuclei, therefore 3C cannot commute with any quantity represented by... 

As we discovered in the last section the spin and. space parts of the eigenfunctions are separately invariant under transformations of the sphere group. The electrostatic forces separate states having different L, and the possible values of S are determined by the Pauli exclusion principle. Although the many-electron Hamiltonian in Eq. 7.18.8 does not contain the spin coordinates—it is therefore invariant under all transformations involving one or more of the electron spin coordinates— the eigenfunctions of 3C do contain spin coordinates. The spin coordinate electron function /(x, y, 2, [Pg.114]

The photon was chosen so that the theory remained invariant under transformations generated by (7 + T3) which for < ,0 would be the usual electromagnetic gauge transformations. To check what happens to the photon under this transformation we must set 0 x) = 0 (x) = 0 and 6 x) = 6 x) in (4.2.3). 

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

In Section 5.2 the set of internal state variables k was introduced. In the referential theory, a similar set of referential internal state variables K will be introduced in the same way without further physical identification at this stage. It will merely be assumed that each member of the set K is invariant under the coordinate transformation (A.50) representing a rigid rotation and translation of the coordinate frame. 

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

Many, possibly all, rules appear to generate asymptotic states which are block-related to configurations evolving according to one of only a small subset of the set of all rules, members of which are left invariant under all block transformations. That is, the infinite time behavior appears to be determined by evolution towards fixed point rule behavior, and the statistical properties of all CA rules can then, in principle, be determined directly from the appropriate block transformations necessary to reach a particular fixed point rule. 

Since the topology of Gf, is invariant under similarity transformations of L, I its complete structure can be computed using L, It remains only to find the cycle sum of a general block C[ ef) ]. We quote the result ([biggs74], [evct79]) ... 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

Note that the scalar product is formally the same as in the nonrela-tivistic case it is, however, now required to be invariant under all orthochronous inhomogeneous Lorentz transformations. The requirement of invariance under orthochronous inhomogeneous Lorentz transformations stems of course from the homogeneity and isotropy of space-time, send corresponds to the assertion that all origins and orientation of the four-dimensional space time manifold are fully equivalent for the description of physical phenomena. 

Now in quantum theory the description of a physical system in the Heisenberg picture for a given observer O is by means of operators Q, which satisfy certain equations of motion and commutation rules with respect to O s frame of reference (coordinate system x). The above notion of an invariance principle can be stated alternatively as follows If, when we change this coordinate frame of reference (i.e., for observer O ) we are able to find a new set of operators that obeys the same equations of motion and the same commutation rules with respect to the new frame of reference (coordinate system x ) we then say that these observers are equivalent and the theory invariant under the transformation x - x. The observable consequences of theory in the new frame (for observer O ) will then clearly be the same as those in the old frame. 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

The theory is, however, invariant under a gauge transformation whereby... 

The above transformation properties of the current operator make quantum electrodynamics invariant under the operation Ue, usually called charge conjugation, provided... 

Then, by virtue of the invariance under proper homogeneous transformations,... 

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