Exam distribution
Rubrics used for the grading
Part 1a (34 points) : Cv or chi (fluctuation dissipation)
100 %: practically perfect
75 %: dimension not correct, otherwise correct
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 1b (34 points) : equipartition theorem: semi-classical, k_B T/2 per quadratic degree of freedom
100 %: practically perfect
80 %: did not mention "(semi-)classical" or "quadratic degree of freedom" but correct otherwise, and gave some valid examples
80 %: did not mention "quadratic" degree of freedom but correct otherwise
60 %: did not mention "(semi-)classical" or "quadratic degree of freedom" but correct otherwise
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 1c (34 points) : Equipartition theorem -- still useful in the quantum regime
100 %: practically perfect
75 %: correct answer, insufficient reason (did not realize that it is useful even in the quantum degenerate regime)
60 %: incorrect answer, but all reasons to give the correct answer are listed
40 %: incorrect answer, but point of view explained well
25 %: incorrect answer, with valid but irrelevant discussion
25 %: correct answer deduced from incorrect reasons!
0 %: answer missing, or completely incorrect
Part 1d (34 points) :
40 %: outline provided, but insufficient calculation to support the answer
33 %: correct answer, but derived simply from E, which is not explained.
0 %: answer missing, or completely incorrect
Part 2a1 (25 points) : exp(beta mu) = n lambda^3, for the volume gas
100 %: practically perfect
60 %: correct starting formula, mistakes lead to incorrect result at end
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 2a2 (25 points) : exp(beta (mu + phi)) = n_A lambda^2, for the surface gas
100 %: practically perfect
80 %: correct except the incorrect power of lambda
50 %: correct starting point, derived an answer, incorrect due to mistake
25 %: incorrect due to incorrect assumptions
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 2a3 (25 points) : equate mu and T (between surface and volume gases) to get n_A = P beta lambda exp(beta phi)
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
33 %: almost correct answer (lambda missing) but due to considering the surface gas only and mis-attributing P of surface to P of bulk
25 %: stated, incorrectly, G_s = G_v, and no result from it
0 %: answer missing, or completely incorrect
Part 2b (25 points) : n_A = 0 when the volume gas is in the BEC phase
25 %: correct starting formula, but no or little development from it
25 %: correct answer deduced from incorrect reasons!
0 %: answer missing, or completely incorrect
Part 3a1 (23 points) : mean field model: H_i -- eigenvalues
100 %: practically perfect
50 %: H_i set, but eigenvalues were not considered
33 %: H_i was misidentified (sigma_x term or sigma_z term not inluded)
0 %: answer missing, or completely incorrect
Part 3a2 (22 points) : Z from H_i eigenvalues
100 %: practically perfect
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 3a3 (23 points) : m_z, m_x from Z, by differentiation
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
90 %: correct except that a sign is incorrect
80 %: logically correct answer from previous incorrect part(s), but with some error (e.g. sign mistake)
50 %: one of m_z, m_x, would have been perfect, were it not for mistakes accumulated from previous part(s)
0 %: answer missing, or completely incorrect
Part 3b (22 points) : h = 0: m_z = tanh (beta J z m_z) like in class, and T_c = J z / k_B
100 %: practically perfect
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
60 %: m_z equation set correctly (or logically correctly from a previously incorrect answer), but T_c incorrectly deduced
25 %: correct answer stated with no derivation
0 %: answer missing, or completely incorrect
Part 3c (10 points) : chi_x = mu_B beta, chi_z = 0
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
50 %: while logically correct from previous incorrect part(s), answers are far from physical
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 4a1 (25 points) : Z for the system set up correctly (including the lid)
100 %: practically perfect
75 %: correct except that p_y was not considered
50 %: did not do the lid, but only gas (and included gravity for the gas, despite the instruction not to)
25 %: correct starting formula, but no or little development from it
Part 4a2 (25 points) : Z integrated correctly?
75 %: expressed as a y-sum, which was not evaluated/integrated.
60 %: y was not integrated over
60 %: integrated the gas part, without/despite noticing that it depends on y
60 %: did not include the volume integral for the gas part
25 %: correct starting formula, but no or little development from it
Part 4b1 (25 points) : -k_B T log Z_{gas} = Gibbs free energy in this case, per principle
100 %: practically perfect
25 %: correct answer deduced from incorrect reasons!
0 %: answer missing, or completely incorrect
Part 4b2 (25 points) : -k_B T log Z_{gas} = Gibbs free energy in this case, per calculation
100 %: practically perfect
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
75 %: discussion is valid, but did not really connect to part a
50 %: tried proving it, in a logically consistent manner from the previous part, but without success
25 %: only general case discussed, not this case, which required examination
0 %: answer missing, or completely incorrect