Momentum eigenfunctions

Here (1/2%) exp(ikx) is the normalized eigenfunction ofF =-ihd/dx corresponding to momentum eigenvalue hk. These momentum eigenfunctions are orthonormal ... 

It may serve to make these results more transparent if we write them out in terms of the familiar momentum eigenfunctions of a particle in a finite box of side L with periodic boundary conditions.12 In this case we have... 

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 ... 

The chemical reaction corresponds to a preparation-registration type of process. With the volume periodic boundary conditions for the momentum eigenfunction, the set of stationary wavefiinctions form a Hilbert space for a system of n-electrons and m-nuclei. All states can be said to exist in the sense that, given the appropriate energy E, if they can be populated, they will be. Observe that the spectra contains all states of the supermolecule besides the colliding subsets. The initial conditions define the reactants, e.g. 1R(P) >. The problem boils down to solving eq.(19) under the boundary conditions defining the characteristics of the experiment. 

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.

In the second line of the equation we have used that Hq, in this one degree of freedom case, contains only kinetic energy in the coordinate rq since the potential energy term is zero in the reactant/product region. In the sixth line we have introduced the momentum eigenfunctions in the coordinate representation, using Eq. (F.23). 

The matrix elements in the integral are evaluated by introducing a unit operator using the momentum eigenfunctions between the operator and the coordinate eigenfunction ... 

In the first line we have introduced the unit operator using the momentum eigenstates because we may then evaluate the resulting matrix elements. In the third line we have introduced the momentum eigenfunctions in the coordinate representation, and in the last line we have used the standard integral f dx exp(— p2x2 qx) = exp(matrix element in Eq. (F.33) we find by analogy that... 

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. 

---