Two-dimensional scattering theory. By analogy with Section 10.2, develop partial wave analysis for two dimensions.
(a) In polar coordinates (r,θ) the Laplacian is

Find the separable solutions to the (time-independent) Schrödinger equation, for a potential with azimuthal symmetry (V(r,θ) → V(r)).

where j is an integer, and u Ξ √r R satisfies the radial equation

(b) By solving the radial equation for very large r (where both V(r) and the centrifugal term go to zero), show that an outgoing radial wave has the asymptotic form

where k Ξ √2mE/ћ. Check that an incident wave of the form Ae^ikx satisfies the Schrödinger equation, for V(r) = 0 (this is trivial, if you use cartesian coordinates). Write down the two-dimensional analog to Equation 10.12, and compare your result to Problem 10.2.

(c) Construct the analog to Equation 10.21 (the wave function in the region where V(r) = 0 but the centrifugal term cannot be ignored).

where H⁽¹⁾ is the Hankel function (not the spherical Hankel function!) of order j.

(d) For large z,

Use this to show that

(e) Adapt the argument of Section 10.1.2 to this two-dimensional geometry. Instead of the area dσ, we have a length, db, and in place of the solid angle dΩ we have the increment of scattering angle |dθ|; the role of the differential cross-section is played by

and the effective “width” of the target (analogous to the total cross-section) is

Show that

(f) Consider the case of scattering from a hard disk (or, in three dimensions, an infinite cylinder) of radius a:

By imposing appropriate boundary conditions at r = a, determine B.
You’ll need the analog to Rayleigh’s formula

(where j_j is the Bessel function of order J). Plot B as a function of ka, for 0 < ka < 2.

Problem 10.2

Construct the analogs to Equation 10.12 for one-dimensional and two-dimensional scattering.

Transcribed Image Text:

D² = a² a² + əx² Əy² = a² 1 a + ar² r är + 1 a² r² 89² (10.105)