Homework Question 1 1. Describe all possible ways in

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Homework Question 1 1. Describe all possible ways in Powered By Docstoc
					                                    Homework

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.


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