0198506961.pdf

230 Magnetic trapping, evaporative cooling and Bose–Einstein condensation

Atom

Atom

(a)

(b)

Incident

plane

wave

Scattered spherical wave

Fig. 10.8(a) A pair of colliding atoms with relative velocityvin their centre-of-mass frame. The impact parameterrimpact
determines their relative orbital angular momentum (which is conserved). (b) In the quantum mechanical description of a
low-energy scattering the solution of the Schr ̈odinger equation is the sum of an incident plane wave eikzplus the wave scattered
by the potential that expands outwards fromr= 0, i.e. a wavefunction of the formψ∝eikz+fk(θ)eikr/r.Thisisaneigenstate
with energyE=
^2 k^2 /M(for elastic scattering, i.e. no loss of energy). Only the scattering amplitudefk(θ) depends on the
potentialV(r). For low energies the scattering amplitude is a constant,f∝Y 0 , 0 , and the condition in eqn 10.20 is fulfilled so
that the phaseskzandkrhave a negligible variation across the region of interaction; henceψ 1+f/r. Writing the scattering
amplitude asf=−aso thatψ(r) = 0 on a spherical surface of radiusr=a, shows that the scattered wave is a spherical wave
that is equivalent to the scattering from a hard sphere of this radius. The comparison with a hard sphere is useful for positive
values ofa, but scattering theory allows negative valuesa<0 for which the outgoing wave is also spherical.

Fig. 10.9 The solution of the
Schr ̈odinger equation for low-energy
scattering from a molecular potential
(cf. Fig. 10.7). The plot showsP(r),
whereR(r)=P(r)/ris the radial
wavefunction. At long range the
overall effect of any potential (with
a finite range) on the scattered wave
is a phase shift (relative to a wave
scattered from a point-like object at
r = 0, shown as a dotted line)—in
this regionP(r)=sin(kr−φ)and
the scattering is indistinguishable from
that of a hard-sphere potential that
gives the same phase shift. Potentials
are characterised by the radiusaof
the equivalent hard-sphere potential.
Radial wavefunctions can be calculated
by the numerical method outlined
in Exercise 4.10, and the extension
of such a computational approach to
study quantum scattering is described
in Greenhow (1990). Figure from
Butcheret al. (1999).

Wavefunction with potential

Wavefunction without potential

0