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Exercises for Chapter 10 245

(b) In thermodynamics the chemical potential is

the energy required to remove a particle from

the systemμ=∂E/∂N,whereEis the total

energy of the system. Show thatE=^57 Nμ.

(10.8) Expansion of a non-interacting condensate

Although experiments are not carried out with

a non-interacting gas it is instructive to consider

what happens whena= 0. In this case the con-

densate has the same size as the ground state

of a quantum harmonic oscillator (for anyN 0 )

and the initial momentum along each direction

can be estimated from the uncertainty principle.

For atoms of the same mass as sodium (but with

a= 0) released from a trap with a radial os-

cillation frequency of 250 Hz and a frequency of

16 Hz for axial motion, estimate roughly the time

of flight at which the cloud is spherical.

(10.9) Excitations of a Bose condensate

The vibrational modes of a condensate can be

viewed as compression waves that form a stand-

ing wave within the condensate; hence these

modes have frequencies of the order of the speed

of sound divided by the size of the condensate

vs/R. Show that this collective motion of the

condensate occurs at a comparable frequency to

the oscillation of individual atoms in the mag-

netic trap.

(10.10) Derivation of the speed of sound
The time-dependent Schr ̈odinger equation for the
wavefunction of an atom in a Bose–Einstein con-
densate in a uniform potential is

i


dψ

dt

=−


^2

2 M

∇^2 ψ+g|ψ|^2 ψ,

where, for simplicity, the potential has been taken

as zero (V= 0). The wavefunctionψ=ψ 0 e−iμt/


satisfies this equation with a chemical potential

μ=g|ψ 0 |^2.

The trial wavefunction with small fluctuations

can be written as

ψ=

[

ψ 0 +uei(kx−ωt)+v∗e−i(kx−ωt)

]

e−iμt/


=ψ 0 e−iμt/
+δψ(t),

where the amplitudes|u|and|v|are small com-

pared to|ψ 0 |.

(a) Show that substituting this function into the

Schr ̈odinger equation and making suitable

approximations leads to the same zeroth-

order approximation for the chemical poten-

tial given above and

i


d

dt

(δψ(t))

=


^2 k^2

2 M

δψ(t)+g|ψ 0 |^22 δψ(t)+gψ^20 δψ∗(t).

(10.47)

(b) Show that equating terms with the same time

dependence leads to two coupled equations

foruandvthat, in matrix form, are

(

k+2g|ψ 0 |^2 −μgψ^20

g(ψ∗ 0 )^2 k+2g|ψ 0 |^2 −μ

)(

u

v

)

=
ω

(

u

−v

)

,

wherek=
^2 k^2 / 2 M.

(c) Hence show thatuandvare solutions of the

matrix equation

(

k+μ−
ωgψ 02

g(ψ∗ 0 )^2 k+μ+
ω

)(

u

v

)

=0.

From the determinant of this matrix, find

the relation between the angular frequency of

the small oscillationsωand the magnitude of

their wavevectork(the dispersion relation).

Show that for low energies this gives the same

expression for the speed of soundω/kfound

in Section 10.7.1.^50

(10.11)Attractive interactions

In certain hyperfine states, the scattering length

aof alkali metal atoms changes with the applied

magnetic field and this feature has been used to

perform experiments in which the atoms have at-

tractive interactionsa<0.

Show that eqn 10.33 can be written in the form

4

3

E


ω

=x−^2 +x^2 +Gx−^3.

By plotting graphs for various values of the pa-

rameterGbetween 0 and−1, estimate the lowest

value ofGfor which there exists a minimum in

the energy as a function ofx. For an atomic

species with a scattering length ofa=−5nm

in a trap whereaho=2μm, estimate the max-

imum number of atoms that a Bose condensate

can contain without collapsing.

(^50) After problem devised by Professor Keith Burnett, Physics graduate class, University of Oxford.