Free and Exact Radial Wave Functions
The present chapter treats the spatial properties of the scattering process as described by the wave function. After an introduction in Section XVII.2 the differential equation for the radial wave function (the Schrödinger equation) is derived. Section XVII.3 presents the solutions of the free radial wave equation, and lists some of their properties. The properties of the exact radial wave functions, in particular their asymptotic forms and their connection to the phase shifts and the S-matrix, are discussed in Section XVII.4. In Section XVII.5 the connection between bound states and poles of the S-matrix on the positive imaginary axis is established. Some material about functions of a complex variable, which is needed in this chapter and in Chapter XVIII, is reviewed in a mathematical appendix.
KeywordsRiemann Surface Branch Point Multivalued Function Radial Wave Function Positive Real Axis
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- 3.For the scattering of electrons by hydrogen atoms this is discussed in Massey et al. ( 1969, Vol. 1, Section 7.1). A more general case is treated in Smith (1971, Section 2. 1 ).Google Scholar
- 12.For potential scattering, the connection between simple poles and bound-state eigenvalues of H can be made more precise, Cf. Taylor ( 1972, Chapter 12).Google Scholar