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Small x:
Note that for small
(
)
is proportional to
and
is proportional to
, which are
the regular and irregular solutions to the separated Laplace
equation. This is the correct way to obtain the static limit.
Large x:
Note that both solutions are regular (go to zero smoothly) at infinity
and are the same (trig) function shifted by
over
there.
Note that they are not square integrable on
(for your
quantum course) but are still better than plane waves in that regard.
Something to think about ...
Next: Hankel Functions
Up: Properties of Spherical Bessel
Previous: The Lowest Few Functions
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Robert G. Brown
2017-07-11