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We can assume, without loss of generality, that the incident wavefunction
is characterized by a wavevector which is aligned parallel to the axis.
The scattered wavefunction is characterized by a wavevector
which has the same magnitude as , but, in general, points
in a different direction. The direction of is specified
by the polar angle (i.e., the angle subtended between the
two wavevectors), and an azimuthal angle about the axis.
Equations (1269) and (1270) strongly suggest that for a spherically symmetric
scattering potential [i.e.,
] the scattering amplitude
is a function of only: i.e.,

(1282) 
It follows that neither the incident wavefunction,

(1283) 
nor the large form of the total wavefunction,

(1284) 
depend on the azimuthal angle .
Outside the range of the scattering potential, both
and
satisfy the free space Schrödinger equation

(1285) 
What is the most general solution to this equation in spherical polar
coordinates which does not depend on the azimuthal angle ?
Separation of variables yields

(1286) 
since the Legendre functions
form a complete
set in space. The Legendre functions are related to the
spherical harmonics, introduced in Cha. 8, via

(1287) 
Equations (1285) and (1286) can be combined to give

(1288) 
The two independent solutions to this equation are the
spherical Bessel functions, and
, introduced in Sect. 9.3.
Recall that
Note that the are wellbehaved in the limit
, whereas the become singular.
The asymptotic behaviour of these functions in the limit
is
We can write

(1293) 
where the are constants. Note there are no functions in
this expression, because they are not wellbehaved as
.
The Legendre functions are orthonormal,

(1294) 
so we can invert the above expansion to give

(1295) 
It is wellknown that

(1296) 
where
[see M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, (Dover, New York NY, 1965),
Eq. 10.1.14]. Thus,

(1297) 
giving

(1298) 
The above expression tells us how to decompose
the incident planewave into
a series of spherical waves. These waves are usually termed ``partial waves''.
The most general expression for the total wavefunction outside the
scattering region is

(1299) 
where the and are constants.
Note that the functions are allowed to appear
in this expansion, because
its region of validity does not include the origin. In the large
limit, the total wavefunction reduces to

(1300) 
where use has been made of Eqs. (1291) and (1292). The above expression can also
be written

(1301) 
where the sine and cosine functions have been combined to give a
sine function which is phaseshifted by . Note that
and
.
Equation (1301) yields

(1302) 
which contains both incoming and outgoing spherical waves. What is the
source of the incoming waves? Obviously, they must be part of
the large asymptotic expansion of the incident wavefunction. In fact,
it is easily seen from Eqs. (1291) and (1298)
that

(1303) 
in the large limit. Now, Eqs. (1283) and (1284) give

(1304) 
Note that the righthand side consists of an outgoing spherical
wave only. This implies that the coefficients of the incoming spherical waves
in the large expansions of and
must be the same. It follows from Eqs. (1302) and (1303) that

(1305) 
Thus, Eqs. (1302)(1304) yield

(1306) 
Clearly, determining the scattering amplitude
via a decomposition into
partial waves (i.e., spherical waves) is equivalent to determining
the phaseshifts .
Now, the differential scattering crosssection
is simply
the modulus squared of the scattering amplitude [see Eq. (1266)]. The
total crosssection is thus given by
where
. It follows that

(1308) 
where use has been made of Eq. (1294).
Next: Determination of PhaseShifts
Up: Scattering Theory
Previous: Born Approximation
Richard Fitzpatrick
20100720