Abstract
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.
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References
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.
We shall return to the many-channel problem in Chapter XX.
For further properties and of the spherical Bessel functions and their proofs see Abramowitz et al. (1965). Chapter 10, and Watson (1958).
See, e.g., Taylor (1972).
Note also that in general any assumed V(r) is a mathematical idealization of the actual situation.
The forces between projectile and target are then also called repulsive and attractive, respectively.
For a review of important results in the theory of functions of a complex variable, see Appendix XVII.A.
The above arguments will also hold for a pole of higher order, but one can prove that these poles are simple (which we shall not, however, do here).
For potential scattering, the connection between simple poles and bound-state eigen-values of H can be made more precise. Cf. Taylor (1972), Chapter 12.
These “redundant” poles may occur if the asymptotic expressions in the complex p-plane differ from (4.17). For interactions that are cut off at a finite distance and also for potentials that fall off at infinity faster than any exponential (e.g. e-μ), “redundant” poles are absent.
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© 1986 Springer Science+Business Media New York
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Bohm, A. (1986). Free and Exact Radial Wave Functions. In: Quantum Mechanics: Foundations and Applications. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-01168-3_17
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DOI: https://doi.org/10.1007/978-3-662-01168-3_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13985-0
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