Radial wavefunction partial-wave

In the partial wave theory free electrons are treated as waves. An electron with momentum k has a wavefunction y(k,r), which is expressed as a linear combination of partial waves, each of which is separable into an angular function Yi (0. ) (a spherical harmonic) and a radial function / L(k,r),... 

The y>Ee(R) are the radial free-state wavefunctions (see Chapter 5 for details). The free state energies E are positive and the bound state energies E(v,S) are negative v and ( are vibrational and rotational dimer quantum numbers t is also the angular momentum quantum number of the fth partial wave. The g( are nuclear weights. We will occasionally refer to a third partition sum, that of pre-dissociating (sometimes called metastable ) dimer states,... 

For dissimilar pairs, the parameter ys equals zero and we have Eq. 5.36. Like pairs of zero spin are bosons and all odd-numbered partial waves are ruled out by the requirement of even wavefunctions of the pair this calls for ys = 1. In general, for like pairs, the symmetry parameter ys will be between -1 and 1, depending on the monomer spins (fermions or bosons) and the various total spin functions of the pair. A simple example is considered below (p. 288ff.). If vibrational states are excited, the radial wavefunctions xp must be obtained from the vibrationally averaged potential, Fq(R). The functions gf(R) and gM(R) are similar to the pair distribution function, namely [294]... 

We begin our discussion with the simple case of a spinless particle of mass m and kinetic energy E = h2k2/2m in a spherical, time-independent potential V(r), so that the Schrodinger equation can be decomposed into uncoupled partial waves l. For a particular l, the scattering matrix or the S matrix is defined as S(k) = exp[2/5(A )] in terms of the phase shift 8(k). Here and in the following, the subscript l on the S matrix and the phase shift is suppressed. The asymptotic form of the time-dependent radial wavefunction is expressible as... 

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