Abstract
In this chapter we derive general results for spherically symmetric situations, i.e., for situations when in addition to the free Hamiltonian K, the interaction Hamiltonian V and the transition operator T also commute with the operator of angular momentum. For the sake of simplicity we consider the spin-zero case. In Section XVI.1, the partial-wave expansion is discussed and the partial cross sections are derived. In Section XVI.2, the phase shift is introduced as a consequence of the unitarity of the S-matrix. In Section XVI.3, a graphical representation of the partial-wave amplitude, the Argand diagram, is introduced.
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References
The meaning of putting the quantum numbers in parentheses has been explained in Section XV.1.
To evaluate the integral one can also use the mathematical formalism for δ(f(x)) developed in Gelfand and Shilov (1964), Vol. 1, Section II.2.5.
Note the change in notation: EA here denotes the total energy, whereas in Section XIV.5 EA denoted the projectile (kinetic) energy.
We have restored here the factor h which had been set equal to unity in the preceding expressions for the cross sections. In (1.21’) the factor on the right-hand side is ℏ. With ℏ = 1 the length units, cm, are identical to the inverse of the momentum units, erg sec/cm or eV/c, and the cross section will have the units of inverse momentum squared instead of area.
We shall suppress the label ηA designating the internal properties of the initial state, whenever this does not obfuscate the notation.
TheS- and T-operators were discussed in more detail in Section XV.3, and (1.41) is identical with (XV.3.36). If Chapter XV was skipped in the first reading, the reader may take (1.41) as the definition of the S-matrix.
For the case that one does not assume a Hamiltonian time development (2.1) is taken as one of the basic postulates for the 5-operator.
The ηl that appears in (2.10) is not to be confused with our use of η to denote the internal quantum numbers. We use it here because this notation of ℏl and ηl is often used in particle physics; in other areas of physics the notation may be different—e.g., in atomic physics ηl is often the symbol for the phase shift denoted by ℏlhere.
ℏb,ℏA denote the internal quantum numbers (channels), tf denotes the inelasticity coefficient for channel ℏb as in Equation (3.4).
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© 1986 Springer Science+Business Media New York
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Bohm, A. (1986). Elastic and Inelastic Scattering for Spherically Symmetric Interactions. In: Quantum Mechanics: Foundations and Applications. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-01168-3_16
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DOI: https://doi.org/10.1007/978-3-662-01168-3_16
Publisher Name: Springer, Berlin, Heidelberg
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