10.5 Bose–Einstein condensation in trapped atomic vapours 229The quantum statistics of identical particles applies to composite parti-
cles in the same way as for elementary particles, so long as the internal
degrees of freedom are not excited. This condition is well satisfied for
cold atoms since the energy required to excite the atomic electrons is
much greater than the interaction energy.^21
(^21) P. Ehrenfest and J. R. Oppenheimer
proved the spin-statistics relation for
indistinguishable composite particles in
‘Note on the statistics of nuclei’,Phys.
Rev., 37 , 333 (1931).
10.5.1 The scattering length
The quantum theory of scattering is well described in most quantum
mechanics texts, and the brief summary given here is only intended as
a reminder of the salient points that are relevant for understanding the
collisions between ultra-cold atoms in a gas. An important feature of
very low-energy collisions is that, although the potential of the attractive
interaction between two atoms has the shape shown in Fig. 10.7, the
overall effect is the same as a hard-sphere potential. Thus we can model
a low-temperature cloud of atoms as a gas of hard spheres,^22 in particular^22 At least in most of the cases of inter-
for the calculation of the contribution to the energy of the gas from est for ultra-cold atoms.
interactions between the atoms.
This section gives a justification for this behaviour of ultra-cold atoms
by simple physical arguments without mathematical details. The molec-
ular potential (as in Fig. 10.7) has bound states that correspond to a
diatomic molecule formed by the two atoms, and this part of the molec-
ular wavefunction is a standing wave analogous to those that lead to
the quantised energy levels of electrons in atoms.^23 It is the unbound^23 This part of the molecular wave-
function represents the vibrational mo-
tion of the molecule. Other aspects
of molecular physics are described in
Atkins (1994).
states, however, that are appropriate for describing collisions between
atoms in a gas and these correspond to travelling-wave solutions of the
Schr ̈odinger equation (illustrated in Figs 10.8 and 10.9). In the quan-
tum mechanical treatment we need to solve the Schr ̈odinger equation to
determine what happens to an atomic wavepacket in the potential. The
angular momentum of a particle in a radial potentialV(r) is a conserved
Fig. 10.7The potentialV(r)forthe
interaction between two neutral atoms
as a function of their separationr.
At small separations the shells of elec-
trons around each atom overlap and the
strong electrostatic repulsion keeps the
atoms apart, while at larger separations
attractive van der Waals interactions
dominate. These forces balance each
other at the separation, where the po-
tential is a minimum—this corresponds
to the equilibrium separation of the di-
atomic molecule that is the bound state
of the two atoms, e.g. the two sodium
atoms are 0.3nmapartinNa 2. This in-
teratomic potential is called the molec-
ular potential and it determines other
properties of the molecule, e.g. its vi-
brational frequency (corresponding to
oscillations in the potential well).