Isotropy of space

In the strictest meaning, the total wave function cannot be separated since there are many kinds of interactions between the nuclear and electronic degrees of freedom (see later). However, for practical purposes, one can separate the total wave function partially or completely, depending on considerations relative to the magnitude of the various interactions. Owing to the uniformity and isotropy of space, the translational and rotational degrees of freedom of an isolated molecule can be described by cyclic coordinates, and can in principle be separated. Note that the separation of the rotational degrees of freedom is not trivial [37]. 

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. 

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows... 

An important corollary of this analogy implies that the conservation of momentum is a consequence of the isotropy of space, whereas energy conservation is dictated by time-inversion symmetry. 

On the other hand, the entity conserved in a closed system due to the isotropy of space is the orbital angular momentum of the system. Apart from a constant factor, the operator fa x Va must therefore correspond to the orbital angular momentum. Further, the angular momentum is an observable (i.e., real valued). Thus the corresponding operator ought to be Hermitian. An operator is said to be Hermitian if it obeys the turn-over rule, that is,... 

Note also that the isotropy of space is the basis for the derivation of the law of conservation of angular momentum [52] [43]. 

In the limit of vanishing anisotropy the quantum numbers j, which describes the rotation of the diatom in the dimer, and /, which corresponds to the rotation of the vector R are good quantum numbers. The total angular momentum J = j +1 is always conserved, due to the isotropy of space, but j and l are broken by the anisotropy in the potential. A degenerate level splits into sublevels J = j — l, ..., j + l under the influence of the anisotropy. If these splittings are small, like in the Ar-Ha (61) and He- H (62) cases, the states can still be labeled to a good approximation by j and 1. 

Stark and Zeeman polarization quantum beats are discussed in Section 6.5.3. An external electric or magnetic field destroys the isotropy of space. As a result, the amplitudes for two transition sequences J", M" — J, M = M" 1 —> J ", M" interfere, and the intensity of X or Y (but not Z) polarized fluorescence is modulated at (Fj M =M"+i — Ejim =M"-i)/h. However, it is not necessary to destroy the isotropy of space in order to observe polarization quantum beats. 

These principles may be regarded as the foundations of science." The homogeneity of time allows the expectation that repeating experiments will give the same results. The homogeneity of space makes it possible to compare the results of the same experiments carried out in two different laboratories. Finally, the isotropy of space allows one to reject any suspicion that a different orientation of our laboratory bench changes the result. 

The conservation laws described are of a fundamental character because they are related to the homogeneity of space and time, the isotropy of space, and the non-distinguishability of identical particles. 

In summary, assumptions about the homogeneity of space and time, isotropy of space, and parity conservation lead to the following quantum numbers (indices) for the spectroscopic states ... 

Its total angular momentum vector remains invariant (because of the isotropy of space). 

The statistical mechanics of the Curie Weiss mean field or the van der Waals mean field can likewise be discussed by the method of random fields (Siegert (1963) Jalickee (1969)). In these cases the mean field analogous to 0(r) is a position-independent vector. The existence of this mean field, however, implies the destruction of the isotropy of space, i.e., the breaking of a symmetry. As Edwards (1970a, b) notes, therefore, there must also be a breaking of symmetry in order to obtain electron localization in the transla-tionally invariant averaged system. 

Isotropy of space (invariance of system constitutive properties with space directions). 

To put this relationship under the form of an impedance, one needs to link with the stress, which can be done easily by assuming the linearity of the elasticity (around a working point) and the isotropy of space, so its Fourier transformation being a scalar, one has... 

We note that (6.12) exhibits translational invariance, due to the isotropy of space, as it must. Thus, if in (6.12) the transformation... 

We could attempt to introduce an SCF approximation directly into (6.16). Such a discussion would be instructive, but only heuristic. The formal derivation is presented and generalized in Section VID. The assumption of the existence of a suitable self-consistcnt field implies that somehow we destroy the isotropy of space. The anisotropy associated with the introduction of an SCF is introduced either by specifying that the initial segment is at some fixed point in space (conveniently chosen as the origin) or by specifying the end-to-end vector R in addition. In the first case, the assumption that r(0) = 0 leads to a polymer distribution which is spherically symmetric about the origin. The field representing the excluded volume then of course has the same symmetry. We want to introduce some approximation that will permit us to calculate both the distribution and the field in a completely self-consistent manner. In the second approach, the specification of r(0) = 0 and r L) = R leads to a field of T>oo7 symmetry about these two end (focal) points. 

In addition to destroying the isotropy of space, the introduction of the SCF field implies a Markovian (albeit self-consistent) approximation to the inherently non-Markovian process described by (6.16). Thus, if EjiCK... 

Because of the assumption of a homogeneous space-time the function B (i>j) cannot depend on the space-time coordinates t and r, and because of the assumption of spatial rotational invariance (isotropy of space) the function must not depend on the direction of but only on its magnitude... 

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