Abstract
A hydrogen atom in the ground state scatters an electron with kinetic energy too small for inelastic collisions to occur. The wave function Ψ(r
1
; r
2
) of the system has boundary conditions at infinity which must be chosen to allow correctly for the possibilities of both direct and exchange scattering. The expansion Ψ = Σ ψ,(r
1
)F
y
(r
2
) of the total wave function in y terms of a complete set of hydrogen atom wave functions ψ
y
(r
1
) includes an integration over the continuous spectrum. It is si own that the integrand contains a singularity. The explicit form of this singularity and its connexion with the boundary conditions are examined in detail. The symmetrized functions Y* may be represented by expansions of the form Σ {ψ
y
(r
1
) G
y
±(r
2
) ±ψ
y
(r
2
) y
G
y
±(r
1
)}, where the integrand in the continuous spectrum does not involve singularities. Finally, it is shown that because all the states ψ
y
of the hydrogen atom are included in the expansion, the equation satisfied by
F
1
, the coefficient of the ground state, contains a polarization potential which behaves like — a/2
r
4
for large r and is independent of the velocity of the incident electron.
Reference11 articles.
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3. Dirac P. A. M. 1947 The principles of quantum mechanics 3rd ed. O xford: Clarendon Press.
4. Theory of Rearrangement Collisions
5. Application of Formal Scattering Theory to Many-Body Problems
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