Angular momentum eigenfunctions

The operators J, J3, and K act only on the angular coordinates of a wave function in polar coordinates. Hence we may consider these expres- 

There is a complete system of common orthonormal eigenvectors  

Hence there is no need to put j as an additional label to the eigenfunctions For each set of eigenvalues ruj and Kj one finds indeed two orthogonal eigenfunctions hence the corresponding common eigenspace 

The functions are spherical harmonics. They form a basis in the set of square-integrable functions on the unit sphere 5. The indices of the spherical harmonics are the numbers = 0,1,2. and m = —i, + For these 

For the sake of completeness, we also provide the definition of the associated Legendre polynomials 

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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... 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

The TD wavefunction satisfying the Schrodinger equation ih d/dt) F(f) = // (/,) can be expanded in a basis set whose elements are the product of the translational basis of R, vibrational wavefunctions for r, r2, and the body-fixed (BF) total angular momentum eigenfunctions as41... 

The parity-adapted total angular momentum eigenfunctions [84] are defined as... 

The term in curly brackets in this equation can be rewritten in terms of total angular momentum eigenfunctions, (R, f), which are obtained by coupling... 

The total angular momentum eigenfunctions, 7(R,f), have parity (—1)- therefore, the summation in Eqs. (A.4) and (A.5) extends over both positive and negative parities. The function is the space-fixed radial scattering... 

In order to transform to the body-fixed representation, we will need to relate the angular functions Wj (R,r) to angular functions defined relative to the body-fixed axes [L., J,K,M,p)QjK ), where J,K,M,p) are the parity-adapted total angular momentum eigenfunctions of Eq. (4.5) and x(0) normalized associated Legendre polynomials of the body-fixed Jacobi angle]. 

Multiplying both sides of this equation by angular momentum eigenfunctions gives ... 

Table 3.2. Angular momentum eigenfunctions in LS-coupling for the electron configuration 2p2. From J. C. Slater, Quantum theory of atomic structure (1960), with the kind permission of J. F. Slater and The McGraw-Hill Companies.

We recall that the square of the total angular momentum J1 commutes with all the components of / and hence with the rotation operator R(rotation operator is applied to an angular momentum eigenfunction j, m), the result is also an eigenfunction of J2 with the same eigenvalue j(j + 1) ... 

Because the Hamiltonian of any central potential quantum system, H p/ commutes with the operators and H, they also have common eigenfunctions, including the situation of confinement by elliptical cones. Although Ref. [8] focused on the hydrogen atom. Ref. [1] included the examples of the free particle confined by elliptical cones with spherical caps, and the harmonic oscillator confined by elliptical cones. They all share the angular momentum eigenfunctions of Eqs. (98-101), which were evaluated in Ref. [8] and could be borrowed immediately. Their radial functions and their... 

In conclusion, the application of the cartesian operators to any angular momentum eigenfunction leads to its companions with the same label i. Then the identification of their spherical or spheroconal representations follow as discussed in Refs. [5] and [6], including their identifications by species and types with their respective numbers depending on whether is even or odd. 

This section is the counterpart of Section 4.1 aimed to illustrate the generation of the complete radial and spheroconal angular momentum eigenfunction for the free particle in three dimensions using an alternative representation of the same operator. 

The generalization by mathematical induction is described next, without including the proof. In fact, the application of the three operators p , Py, on the 2 + 1 linearly independent cartesian angular momentum eigenfunctions multiplied by the radial Bessel function Zt ikr) lead to the following results for even and odd, respectively. 

Similarly, of the 3(4n + 3) combinations for = 2n + l, there are 2(2n + 2) +1 with = 2n + 2 and 2(2n) +1 with = 2n linearly independent combinations, which can be identified with cartesian angular momentum eigenfunctions of even degrees 2n + 2 and 2n, respectively, multiplied by their respective 2a,+2(A r) and Z2n(kr) radial functions. The combinations for the latter are also identified through their additional common factor + y + 7 =... 

In a certain respect the einalysis of the rotational Zeeman effect may appear simpler in the case of symmetric top molecules as compared to the asymmetric top case, since the angular momentum eigenfunctions with K... 

Separation of the variables to enable solution of the electronic wave equation of hJ requires clamping of the nuclei and hence imposing cylindrical symmetry on die system. The calculated angular momentum eigenfunctions are artefacts of diis approximation and do not reflect die full symmetry of die quantum-mechanical molecule. 

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