We can assume, without loss of generality, that the incident wavefunction is characterized by a wavevector that is aligned parallel to the -axis. The scattered wavefunction is characterized by a wavevector that 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. Equation (943) strongly suggests that for a spherically symmetric scattering potential [i.e., ] the scattering amplitude is a function of only: i.e.,
It follows that neither the incident wavefunction,
nor the total wavefunction,
depend on the azimuthal angle .
Outside the range of the scattering potential, both and satisfy the free space Schrödinger equation
Consider the most general solution to this equation in spherical polar coordinates that does not depend on the azimuthal angle . Separation of variables yields
since the Legendre polynomials form a complete set in -space. The Legendre polynomials are related to the spherical harmonics introduced in Chapter 4 via
Equations (958) and (959) can be combined to give
The two independent solutions to this equation are the spherical Bessel function, , and the Neumann function, , where
Note that spherical Bessel functions are well-behaved in the limit , whereas Neumann functions become singular. The asymptotic behaviour of these functions in the limit is
We can write
where the are constants. Note there are no Neumann functions in this expansion, because they are not well-behaved as . The Legendre polynomials are orthogonal,
so we can invert the above expansion to give
It is well-known that
where . Thus,
The above expression tells us how to decompose a plane wave into a series of spherical waves (or ``partial waves'').
The most general solution for the total wavefunction outside the scattering region is
where the and are constants. Note that the Neumann 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
where use has been made of Equations (964)-(965). The above expression can also be written
where the sine and cosine functions have been combined to give a sine function that is phase-shifted by .
Equation (974) yields
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 that
in the large- limit. Now, Equations (956) and (957) give
Note that the right-hand side consists only of an outgoing spherical wave. This implies that the coefficients of the incoming spherical waves in the large- expansions of and must be equal. It follows from Equations (975) and (976) that
Thus, Equations (975)-(977) yield
Clearly, determining the scattering amplitude via a decomposition into partial waves (i.e., spherical waves) is equivalent to determining the phase-shifts .
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)