Eigenfunctions scalar product

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... 

With the choice a = 0, the total eigenfunction xp io first order is normalized. To show this, we form the scalar product xp xp ) using equation (9.29) and retain only zero-order and first-order terms to obtain... 

We conclude that the eigenfunctions are complete. Moreover it follows that the eigenvalues are real and that any two eigenvectors are orthogonal in terms of the scalar product (7.4). For discrete eigenvalues they may be normalized so that... 

In a similar manner we can show that the generators T, and T3 are Hermitian with respect to the same scalar product, Eq. (79). Only the generator T3 has square integrable eigenfunctions and this is the reason why we chose to diagonalize T2 and T3 in our study of the representation theory of so(2, 1) (the notation J2, J3 was used in Section III). 

The presence of an inseparable scalar product ij Sj of one operator that acts only on the spatial part of the total electronic wave function and another one that acts only on its spin part will cause an interaction between the various pure multiplicity wave functions. The resulting eigenfunctions of / ei will therefore be represented by mixtures of functions that differ in multiplicity. However, since constitutes only a very small term in mixing is normally not very severe, and the resulting wave functions contain predominantly functions of only one multiplicity. Commonly, such impure singlets, impure triplets, etc., are still referred to simply as singlets, trip-... 

Taking the scalar product with and using the orthonormality of the eigenfunctions. 

For most of the observables the formulas derived and used below most often will require as input, scalar products of various functions F with eigenfunctions of the Hamiltonian in the box, (F

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Since the eigenfunction n) is not in the Hermitian domain of the Hamiltonian the definition of the inner product that we should use should be questioned. If we will keep the usual definition of the scalar product in quantum mechanics the coefficients an in Eq. 33 will get real positive values only (as well as a(e) in Eq. 32) and the possibility of interference among different resonance states which leads to the trapping of an electron due to the molecular vibrations will be eliminated. As was mentioned before the generalized definition of the inner product (.... ..) rather than the usual scalar product has to be used since the Hamiltonian is... 

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