8.08 Problem Set # 7Problems:1. By repeating the argument that leads to the Bose-Einstein condensation in 3D, show thatthe transition temperature approach zero in an ideal boson gas of particles in 2D.2. If the mechanism for photon absorption or emission can be neglected as may happen in somecosmological settings, the number of photons would be conserved. Can a photon gas undergoBose-Einstein condensation under there circumstance? If so, give the critical photon densityas temperatures T .3. The interference of matter wavesThe photograph above shows the total density of two Bose-Einstein condensates of Na atomswhich move towards each other (upward and downward) and start to overlap. The interferencefringes have a space of 1.5 × 10−5cm. Find the relative velocity between the two condensates.4. density profiles of Bose-Einstein condensation1Consider a gas of bosonic sodium atoms confined in a quadratic potential well U(r) =12mω20|r|2where m is the mass of the sodium atom. The characteristic length of the os-cillator potential is r0=p~/mω0= 5 × 10−3cm.(a) Ignore the interaction between the sodium atoms, find the size of the condensed sodiumatoms at T = 0. How does the size of the condensation depends on the number of particles?(b) For interacting bosons, the shape of condensation at T = 0 is determined byh−~22m∂2r+ (U(r) − µ) + g|ψ(r)|2iψ(r) = 0In Thomas-Fermi approximation, we assume the wave function ψ is smooth and drop the ∂2rterm. In this case the shape of condensation is determined by[(U(r) − µ) + g|ψ(r)|2]ψ(r) = 0Now, how does the size of the condensation depends on the number of particles?(c) In the above, the left picture shows measured shapes of the condensation of Na atoms.The maximum density is 1011cm−3for the shape near T = 0 (labeled by T << Tc). Thewidth of the peak at the base is 0.03cm. Find the interaction strength g and the scatteringlength a of the sodium atom. (Note g and a are related by g