= 7 2. Consider a gas of N spinless non-relativisitic bosons trapped in a 2-dimensional harmonic potential V = {mw? (x2

[answerhappygod]

7 2 Consider A Gas Of N Spinless Non Relativisitic Bosons Trapped In A 2 Dimensional Harmonic Potential V Mw X2 1 (177.98 KiB) Viewed 62 times

= 7 2. Consider a gas of N spinless non-relativisitic bosons trapped in a 2-dimensional harmonic potential V = {mw? (x2 + y2). The single-particle energy levels are (nx + ny +1)ħw. (Laboratory BECs are produced in traps.) a) Optional By considering the number of states with energy between ε and € + de, show that in the continuum approximation, the density of states in energy is (€ – ħw) g(8) for ε > ħw. (h )2 [Hint: we are working directly with energies here, the momentum or wave number do not play a role in the harmonic oscillator. It may help to consider a 2D grid of energy states (rather than as before, momentum states).] b) Not optional Using the density of states above, show that for sufficiently large temperatures the relation between the particle number and the chemical potential is kpT N dx, hw 1 where z e[wu)B. (Use the substitution x = (€ – ħw)B.) Hence show that Bose Einstein condensation can occur in this system, and find the critical temperature. 2 dx You should be able to argue that the integral is finite even without 1 6 knowing that though.] 2 х = (. zex roo х [Hint: 6 5-1 = ex