1. Describe all possible ways in which you can you arrange Ns = 3 bosons in
Ms = 4 quantum states.
2. In how many ways can you arrange Ns = 3 distinguishable Maxwell–
Boltzmann particles in Ms = 4 states ? (just state the number)
A homogeneous condensate is described by the following Gross–Pitaevskii
equation for the complex wavefunction ψ(x, y, z, t):
∂ψ ¯2 2
h =− ∇ ψ + V0 |ψ|2 ψ − µψ,
where µ is the chemical potential, V0 is the interaction potential, and the nor-
malization condition is
∞ ∞ ∞
|ψ|2 dx dy dz = N.
−∞ −∞ −∞
1. Determine the dimensions of each term of the equation. Hint: the dimen-
sions of h, V0 and µ are respectively [¯ ] = erg s (where erg = g cm2 /s2 ),
[V0 ] = g cm5 /s2 and [µ] = erg.
2. Assume that ψ is independent of x, y, z, t and that it is real, hence show
that the solution which corresponds to a uniform condensate away from
boundaries is ψ∞ = µ/V0 .
3. Consider the GP equation. Balance the kinetic energy term (¯ 2 /(2m))∇2 ψ
and the interaction term V0 |ψ|2 ψ at some scale ξ, hence show that the scale
ξ= √ .
4. Rewrite the Gross–Pitaeviski equation in dimensionless form, using ξ as
length scale, τ = h/µ as time scale, and ψ∞ as unit of ψ.
5. Assume that ψ depends only on x. The resulting GP equation is
¯ 2 d2 ψ
0=− + V0 |ψ|2 ψ − µψ,
Consider this equation in in the region 0 ≤ x ≤ ∞. Assume that ψ is real
and that ψ(0) = 0 and ψ → ψ∞ for x → ∞. Show that the solution of
the equation is ψ(x) = A tanh (αx), hence determine A and α.