BEC

Question 1

Ideal Bose Gas

Answer 1

Question 2

definition of quantum degeneracy

Answer 2

Question 3

Condition for quantum degeneracy

Answer 3

Question 4

Why BEC? (Why do fermions not show this?)

Answer 4

Question 5

What is BEC?

Answer 5

Question 6

Distribution function for bosons

Answer 6

Question 7

i) ‘particle in a box’ states
1. Find the single particle quantum states

Solve the SE for the eigenstates, find energies and wavefunctions

Answer 7

Question 8

Derive energy density of states g(ε) for a free ideal gas

Answer 8

Question 9

i) ‘particle in a box’ states
2. “counting” the states

Answer 9

no. of states with energy ε -> ε + dε

derive/use energy density of states to find g(ε)dε

Question 10

Adding in ground state by hand to N

Answer 10

At T=0 all particles are in GS but g(ε) ∝ ε^1/2, leaving GS zero contribution to mere integration of nbar(ε)g(ε)dε.

Question 11

μ for macroscopic occupation of the ground state

Answer 11

Question 12

i) ‘particle in a box’ states
3. Occupy the states

Answer 12

n(ε)d(ε) = no. of particles with energy ε -> ε + dε

how are particles distrubted over energy levels? Use B-E dist func nbar(ε) *g(ε)dε = n(ε)dε. Integrate for N.

Add in GS by hand.

mu for macroscopic population

Question 13

ii) ‘BEC in harmonic trap’

Answer 13

Question 14

ii) BEC in harm trap

Answer 14

Question 15

ii) BEC in harm trap

Answer 15

no. of quantum states is volume (no. of states) x f (degrees of freedom)

Question 16

ii) BEC in harm trap

Answer 16

Question 17

ii) BEC in harm trap

Answer 17

Question 18

heat capacity (const volume)

Answer 18

Question 19

Answer 19

Use expression for condensate fraction, rewrite for Nex, sub into internal energy estimate (N* the typ energy kT). Find Cv. Compare to classical C.

Question 20

Evaluate internal energy

Answer 20

previously estimated. Now do integral!