1. This problem comes from D. Schroeder, An introduction to Thermal physics. prob.7.66

Consider a collection of 10,000 atoms of 87Rb confined in a box of volume (10-5m)3

(a) Calculate 0 the energy of the ground state.

(b) Calculate the condensation temperature and compare kBTc to 0

(c) Suppose that the T = 0.9Tc. How many atoms are in the ground state?

How close is the chemical potential to the ground state energy?

How many atoms are in each of the (threefold degenerate) first excited state?

(d) Repeat (b) and (c) for 106 atoms confined to the same volume. Under which conditions

will the atoms in the ground state be much greater than the first excited state?

2. Consider a gas of 23Na atoms confined in a 3D isotopic harmonic potential where,

U(r) = m2r2 where m is the mass of the atom.

(a) Calculate the size of the harmonic oscillator l0 at T=0. Then ignoring interactions

calculate the condensate size. Does the size depend on the atom number N.

(b) The shape of the condensate at T = 0 and with repulsive interactions g > 0 is described by

the Gross-Pitaevskii equation.

State the Thomas Fermi approximation and apply it to the Gross Piteavskii equation given

above. Using this solution derive the equation for the Thomas Fermi Radii and show how it

depends on N.

(c) Calculate the interaction strength and the scattering length, a, for 23Na given that for a

pure Condensate the width at the base is 0.03 cm, and the density 1011 cm3.

3. In the lecture we considered the formation of a non-interacting Bose-Einstein Condensate in a

3D infinite box potential. Now we want to consider what will happen in both 2D and 1D for

the non-interacting case.

(a) Derive the density of states for 1D and 2D.

(b) Using the results derived in (a) show whether or not BEC can be achieved in 1D and

2D.