0198506961.pdf

232 Magnetic trapping, evaporative cooling and Bose–Einstein condensation

example we shall examine the properties of wavefunctions withl=0

(corresponding to s-waves), so that the equation forP(r)issimply

[

−

^2

2 M′

d^2

dr^2

+V(r)

]

P(r)=EP(r), (10.21)

whereM′is the mass of the particle. For the potentialV(r)=0within

the rangea
r
bandV(r)=∞elsewhere (forr<aandr>b),

this equation has the same form as that for an infinite square well in one

dimension. The solution that satisfies the boundary conditionψ(r)=0

atr=ais

P=Csin (k(r−a)), (10.22)

whereC is an arbitrary constant. The boundary condition that the

wavefunction is zero atr=brequires thatk(b−a)=nπ,wherenis an

(^29) This is the usual result for an infinite integer; hence the energy eigenvalues are given by 29
square well of lengthLrewritten with
L=b−a.
E=

^2 k^2
2 M′

=

^2 π^2 n^2

2 M′(b−a)^2

. (10.23)

To make a link with scattering theory we consider what happens when

ab,so that the wavefunction is contained in a spherical region of

radiusbbut excluded from a small hard sphere of radiusaat the origin;

the energy of the lowest level (n= 1) can be written as

E=

^2 π^2

2 M′b^2

(

1 −

a

b

)− 2

E(a=0)+

^2 π^2 a

M′b^3

. (10.24)

This equals the energy fora= 0 plus a small perturbation proportional

toawhich arises because the kinetic energy depends on the size of the

region betweenr=aandb. (The expectation value of the potential

energy is zero.) At short range where sin (k(r−a))k(r−a)the

solution in eqn 10.22 reduces to

R(r)

P(r)

r

∝ 1 −

a

r

. (10.25)

This is the general form for a wavefunction with a low energy (ka1)

near a hard sphere (in the regiona<rλdB/ 2 π) and the features

illustrated by this example arise in the general case. The use of this

situation to illustrate scattering might seem to contradict the above

assertion that scattering involves wavefunctions that are unbound states,

but we will see below that such waves have a similar increase in energy

proportional toa. In any case, we will be applying the results to atoms

that are confined (in the harmonic potential created by a magnetic trap).

A collision between a pair of atoms is described in their centre-of-

mass frame as the scattering from a potentialV(r) of a particle with a

reduced mass given by^30

(^30) The reduced mass is used for the
same reason as in the hydrogen atom,
i.e. in one-electron atoms the electron
and nucleus orbit around their centre-
of-mass frame, and therefore it is the
reduced mass of the electronme×
MN/(me+MN) that appears in the
Schr ̈odinger equation for the atom—the
(slight) dependence of the electron’s re-
duced mass on the nuclear massMN
leads to the isotope shift of spectral
lines (Chapter 6).

M′=

M 1 M 2

M 1 +M 2

. (10.26)