Orthogonal property

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... 

Upon inverting Eq. (10-200) by making use of the orthogonality properties of plane wave solutions of the Dirac equation, we obtain the result that... 

E (for the identity) in Table 6 are accounted for. Furthermore, the totally symmetric representation is r(1) e A the latter notation is dial usually used by speetroscopists The construction of the remainder of the character table is accomplished by application of the orthogonality property of the characters [see Eq. (30) and problem 131. Standard character tables have been derived in this way for the more common groups, as given in Appendix VQI. 

This set of simultaneous equations [Eq. (34)] can be solved with aid of the orthogonality properties of the characters, as given by Eq. (30). Multiplication... 

Note that because the effective Hamiltonian matrix is not Hermitian, the eigenvectors are not orthogonal. However, when ac is small, the orthogonality properties are satisfactorily verified. 

On substituting Eq. (5.43) into Eq. (5.41) we have, using the orthogonality properties of the circular functions,... 

The advantage of these functions is their orthogonality property, giving a diagonal matrix for S with 0 or 1 as elements. With this form of S, we have a standard eigenvalue algebraic problem where C and A must be determined. 

The powers of the variable x (1, x, x2,..., xa,...) are not orthogonal functions over a unique interval. However, particular sets of polynomials present the orthogonality property. A simple and useful example is that of Legendre polynomials. Let us choose over the range [—1, +1] the first two polynomials... 

Alternative definitions lack the (— l)m term. P,(x) is therefore a concise expression for P,°(x). Note that, because of the derivative term in equation (2.6.14), if m > l, P,m(x) = 0. Advanced calculus would show orthogonality properties with... 

It is a direct result of the orthogonality property of linear least squares (Section III,A,1) that the crude sum of squares may be broken into two parts as follows ... 

The orthogonal properties of fluoroalkyl amino acid side chains 747... 

The a-helical coiled coil-based screening system already provided a wide variety of information about the interactions of fluorinated amino acids within hydrophobic and hydrophilic protein environments. Investigations on the thermal stability as well as the replicase activity have both emphasized the orthogonal properties of fluorinated aliphatic amino acid side chains. The term orthogonal in this context has been chosen by us to demonstrate that they are in fact hydrophobic... 

Similarly, (Now ,NewM), (Mom ,Now ) and (M N J are mutually orthogonal sets of functions. The orthogonality properties of cos m< > and sin m imply that all vector harmonics of different order m are mutually orthogonal. 

With knowledge of the full OPDF in Eq. (6), the averages in Eq. (8) can be evaluated by integration with respect to the angles f and , which is straightforward due to the orthogonality properties of the spherical harmonics.46... 

The product Y2m> Ylm" can be represented by a summation of spherical harmonics with no terms having /> 4. Thus the orthogonality property of Yim makes all terms in Eq. (24) with /> 4 give no contribution to VMM>. Therefore, we can truncate the expansion for Vc at the 1=4 terms. For rare-earth ions with / electrons, the expansion must be truncated at the 1=6 terms. 

All terms on the left vanish except that for n = due to orthogonality property of wave functions and from the conditions of normalization,... 

Considering the orthogonality properties of the cyclic unit vectors ... 

After using a circular permutation of the bra and kets that do not belong to the spaces where the exponential operators work and then, after simplification using the orthogonality property ( 0) = 8 it reads... 

Introducing the expansion [25] into the Fokker-Plank equation [24] and using the orthogonal properties of the Hermite polynomials, the following moment equations are obtained ... 

The 3-j symbol also has the advantage that the projection quantum numbers are shown below their associated vectors, consistent with their subordinate role. The orthogonality properties of the 3-j symbols are written as... 

Equation (7.17) follows from the previous equation because of the orthogonality properties of the basis set, given by equation (7.4). In our subsequent discussion we shall also require the results ... 

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