234 Magnetic trapping, evaporative cooling and Bose–Einstein condensation
Fig. 10.10The wavefunctionP(r)=
rR(r) for very-low-energy scattering
from slightly different molecular poten-
tials with scattering lengths that are
(a) positive and (b) negative with very
large magnitude. From Butcheret al.
(1999). The extrapolation of the wave-
function from at largeris drawn as a
dashed line that crosses the horizontal
axis atr=a.
(a)(b)0010.6 A Bose–Einstein condensate
The interaction between atoms is taken into account by including a term
in the Schr ̈odinger equation that comes from eqn 10.27, proportional to
the square of the wavefunction:
{
−^2
2 M
∇^2 +V(r)+g|ψ|^2}
ψ=μψ. (10.29)The extra energy from the interactions is proportional to|ψ|^2 ,theprob-
ability of finding a particle in a given region, and the coupling constant
is
g=4 π^2 Na
M. (10.30)
This comes from eqn 10.27 with|χ|^2 →N|ψ|^2 that gives the interaction
per atom in the presence ofNatoms.^36 The symbolμis used (instead(^36) Actually, there are N−1other
atoms, but the difference fromNis neg-
ligible for large numbers of atoms.
ofE) to represent the energy of an individual atom in the presence of
all the others (cf. the central-field approximation in Chapter 4).^37 This
(^37) This quantity turns out to be equiv-
alent to the chemical potential in ther-
modynamics: μ=∂E/∂Nis the en-
ergy required to remove a particle from
the system. This is not the same as the
average energy per particle—see Exer-
cise 10.7.
nonlinear Schr ̈odinger equation is called the Gross–Pitaevskii equation
after the people who first (independently) wrote it down and a rigorous