# Homework Question 1 1. Describe all possible ways in

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Question 1

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)

Question 2

A homogeneous condensate is described by the following Gross–Pitaevskii
equation for the complex wavefunction ψ(x, y, z, t):
∂ψ      ¯2 2
h
i¯
h    =−       ∇ ψ + V0 |ψ|2 ψ − µψ,
∂t      2m
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 ),
¯                            h
[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
h
and the interaction term V0 |ψ|2 ψ at some scale ξ, hence show that the scale
ξ is

h
¯
ξ= √    .
2mµ

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
h
0=−               + V0 |ψ|2 ψ − µψ,
2m dx2
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.

1

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