Question

trying for last time :( Can anyone please help and explain how to do this task ? Thank you

Q4 (QUANTUM IDEAL GASES) Is the statement "Given a two-spinless-fermion system, and two orbitals o labeled by quantum numbers a, b, the two-body wavefunction (1,2 represent the particle variables) V (1, 2) = 0a(1)$a (2) - 06(1)$6(2) +0a(1)º(2) - 06(1)$a(2) correctly describes a possible state of the system” true or false ? Explain your answer (0.5p). 4b) Consider a Fermi gas at small temperatures, T «TF. Explain qualitatively why the heat capacity is linear in T. (2.5p) 4c) PROBLEM (3.0p) Two particles are such that each of them can occupy 3 different levels with energies 0, € and 2€. (a) Determine the partition function Z when the particles are classical and distinguishable. (b) Determine Z when the particles are indistinguishable spinless fermions. (c) Find the energy of the system in (b) when T +0. (d) Find the entropy of the system in (b) when T +. (e) Determine Z when the particles are indistinguishable bosons. (f) Find the entropy of the system in (e) when T +0. Hint. To simplify the formulas, use x = exp(-€/kpT).

Answer 1

4a) since particles are spinless fermious, it must obey pauli's exclusion principle - prevents occupancy same tuo fernions on the geven state. In the given care function, the same two fermious occupied either in fe or the states. This is forbiddem, hence geven uare function is not allowed for the given situation. Not possible

4b) a) If wis the total number of fermions, only a fraction of the order of ke/G. Can be enceted thermally at temperature T. (E = Femienezzy b Each of these Neilor fermions has a thermal enessy of the order of KRT. is, the total thermal enersy (w of fermions is, UxNkxI) KAST c) : electric (fermonic) heat Capacity Clem - du x (NEA) KRST In other words, proportional to temperature fermanic T. heat Capacity is 49 a) 1 particles are closial delignable. = = = = F-36 E-26 E= 4€ ERE E-RE LL (doubly 1 (Doubly) (doubly deganate) degemach

The fastition function . zosebe akene pa yunt 2=lt eeße tetke ae aperta e Betz espe = 1+ 3 e2Bet a espeta eßet e FBE. B (6) Espinless fermione B-- = Le 07 E-SE E -E E-26 Lo ēBET -25€ -35€ (c) The energy, U = -2 logz =-2609 @mez PET & PC) a telemetz@&selle) B 2BE + OBBE when T 4x, p=0. :: U= € (1+2+3) - 3€ 1+1+1

e a = op =-kB 8 . -2RE - 3 telt E -28€ -3B6) +2e +3e (d) Entropy, 3 3 3 3 3 Sa-2f-kst log2) =knlogz + kBT 2 loge = kaplogz - Ket 8 Bologz = 4 lg 2-logz. = kz logz + 4 So ke lay (epet etetéte) +eCetizatisch Te kete284+ēBEJ when Top-o : Sa kas log 3 e) when pasticles are bosons (Any munkee can occupy any slate) E Eco E= 2€ Est Ece Erze E3€. Z= teppe te the eße 22 te ape entropy, s= kn logz tut ahen Tods =kn logz Irad - Kelog (1+1+1+1+1+) =ty laga - - - :