Fermi Distribution Function. n(e) T = 0 T > 0 E F

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1 LECTURE 3 Maxwell{Boltzmann, Fermi, and Boe Statitic Suppoe we have a ga of N identical point particle in a box ofvolume V. When we ay \ga", we mean that the particle are not interacting with one another. Suppoe we know the ingle particle tate in thi ga. We would like toknow what are the poible tate of the ytem a a whole. There are 3 poible cae. Which one i appropriate depend on whether we ue Maxwell{Boltzmann, Fermi Boe tatitic. Let' conider a very imple cae in which we have 2 particle in the box and the box ha 2 ingle particle tate. How many ditinct way can we put the particle into the 2 tate? Maxwell{Boltzmann Statitic: Thi i ometime called the claical cae. In thi cae the particle are ditinguihable o let' label them and B. Let' call the 2 ingle particle tate and 2. F Maxwell{Boltzmann tatitic any number of particle can be in any tate. So let' enumerate the tate of the ytem: Single Particle State B B B B We get a total of 4 tate of the ytem a a whole. Half of the tate have the particle bunched in the ame tate and half have them in eparate tate. Boe{Eintein Statitic: Thi i a quantum mechanical cae. Thi mean that the particle are inditinguihable. Both particle are labelled. Recall that boon have integer pin: 0,, 2, etc. F Boe tatitic any number of particle can be in one tate. So let' again enumerate the tate of the ytem: Single Particle State We get a total of 3 tate of the ytem a a whole. 2/3 of the tate have the particle bunched in the ame tate and /3 of the tate have them in eparate tate. Fermi Statitic: Thi i another quantum mechanical cae. gain the particle are inditinguihable. Both particle are labelled. Recall that fermion have half{integer pin: /2, 3/2, etc. ccding to the Pauli excluion principle, no me than one particle can be in any one ingle particle tate. So let' again enumerate the tate of the ytem: Single Particle State

2 We get a total of tate of the ytem a a whole. None of the tate have the particle bunched up; the Pauli excluion principle fbid that. 00% of the tate have the particle in eparate tate. Thi imple example how how thetype of tatitic inuence the poible tate of the ytem. Ditribution Function We can fmalize thi omewhat. We conider a ga of N identical particle in a volume V in equilibrium at the temperature T. We hall ue the following notation: Label the poible quantum tate of a ingle particle by r. Denote the energy of a particle in tate r by " r. Denote the number of particle in tate r by n r. Label the poible quantum tate of the whole ga by R. Since the particle in the ga are not interacting are interacting weakly, we can decribe the tate R of the ytem a having n particle in tate r =,n 2 particle in tate r =2, etc. The total energy of the tate i E R = n " + n 2 " 2 + n 3 " 3 ::: = r n r " r () Since the total number of particle i N, then we mut have The partition function i given by r n r = N (2) Z = R e,e R = R e,(n "+n2"2+:::) (3) Here the um i over all the poible tate R of the whole ga, i.e., eentially over all the variou poible value of the number n, n 2, n 3,... Now we want to nd the mean number <n > of particle in a tate. The <:::> refer to a thermal average. Since P R = e,(n "+n2"2+:::) Z i the probability of nding the ga in a particular tate where there are n particle in tate, n 2 particle in tate 2, etc., one can write f the mean number of particle in a tate : PR n e,(n "+n2"2+:::) <n >= n P R = (5) Z R (4) 2

3 We can rewrite thi a <n >= Z e,(n "+n2"2+:::) <n ln (7) So to calculate the mean number of particle in a given ingle{particle tate, we jut have to calculate the partition function Z and take the appropriate derivative. We want to calculate <n > f both Boe and Fermi tatitic. Boe{Eintein and Photon Statitic Here the particle are to be conidered a inditinguihable, o that the tate of the gacanbepeciedby merely liting the number of particle in each ingle particle tate: n, n 2, n 3,... Since there i no limit to the number of particle that can occupy atate, n can equal 0,,2,3,... f each tate. F photon the total number of particle i not xed ince photon can readily be emitted abbed by thewall of the container. Let' calculate <n > f the cae of photon tatitic. The partition function i given by Z = e,(n "+n2"2+:::) (8) R where the ummation i over all value n r =0; ; 2; 3;::: f each r, without any further retriction. We can rewrite (8) a Z = n=0 Z = n;n2;::: e,n " e,n " e,n 2"2 e,n 3"3 ::: (9) n2=0 e,n 2"2 n3=0 e,n 3"3 ::: (0) But each um i a geometric erie whoe rt term i and where the ratio between ucceive term i exp(," r ). Thu it can be eaily ummed: n =0 Hence eq. (0) become e,n" =+e," + e,2" + ::: = Z =, e,", e,"2, e,"3," ln Z =, ln, e, e," () ::: (2) (3) 3

4 So if we plug thi into eqn. (7), we get <n ln = ln, e," = e,", e," (4) <n >= (5) e ", Thi i called the \Planck ditribution." We'll come back to thi a bit later when we talk about black body radiation. Photon are boon, but their total number i not conerved becaue they can be abbed and emitted. Other type of boon, however, do have their total number conerved. One example i 4 He atom. 4 He atom i a boon becaue if you add the pin of the proton, neutron, and 2 electron, you alway will get an integer. If the number of boon i conerved, then <n > mut atify the condition <n >= N (6) where N i the total number of boon in the ytem. In der to atify thi condition, one lightly modie the Planck ditribution. The reult i known a the Boe{Eintein ditribution <n >= (7) e (",), where i the chemical potential. i adjuted o that eq. (6) i atied. Phyically i the change in the energy of the ytem when one particle i added. Eqn. (7) i called the Boe{Eintein ditribution function the Boe ditribution function f ht. We will return to the Boe{Eintein ditribution when we dicu Boe{Eintein condenation. Fermi{Dirac Statitic Recall that fermion have half{integer pin tatitic and that at mot one fermion could occupy a each ingle particle tate. Thi mean that n = 0. We can eaily get ome idea of what <n > by conidering the very imple cae of a ytem with jut one ingle particle tate. In thi cae <n >= Pn n e,n" Pn e,n " (8) In thi cae the um jut have 2 term. The denominat i The numerat i n =0; n =0; e,n" =+e," (9) n e,n" =0+e," (20) 4

5 So we have <n >= e," +e," (2) <n >= (22) e " + F a real ytem we have many ingle particle tate and many particle. The expreion f < n > in thi cae mut atify the condition that the number of particle i a contant: <n >= N (23) The crect fmula which atie thi condition (23) i <n >= e (",) + Thi i called the Fermi ditribution function. i adjuted to atify the contraint (23). in the Boe{Eintein cae, i called the chemical potential. Thi i baically the ame a the Fermi energy. We will return to thi when we dicu metal and uperconduct. n(e) Fermi Ditribution Function T = 0 (24) T > 0 E F E Claical Limit We can ummarize our reult f the quantum tatitic of ideal gae with <n >= e (",) where the upper ign refer to Fermi tatitic and the lower ign refer to Boe tatitic. If the ga conit of a xed number of particle, i determined by <n >= (25) e (",) = N (26) 5

6 In general the number N of particle i much maller than the total number of ingle particle tate. Let u conider 2 limiting cae. Conider the low denity limit where N i very mall. The relation (26) can then only be atied if each termintheumover all tate i uciently mall, i.e., if <n > exp[(", )] fall tate. The other cae to conider i the high temperature limit. Since ==k B T,thehigh temperature limit crepond to mall. Now if were 0, we would have = N (27) which i a diater f both the Fermi{Dirac and Boe{Eintein cae. But = 0 mean that T =. Let' aume that the temperature i high but not innite, o that i mall but not 0. t high temperature, lot of high energy tate are occupied. By \high energy," I mean that ". In der to atify the xed N contraint of eqn. (26), it i neceary to have exp[(", )] (28) uch that <n > (29) f all tate. (Remember that there are many me tate than particle N.) Thi i the ame condition that came up in the low denity cae. We call the limit of uciently low concentration uciently high temperature where (28) (29) are atied the \claical limit." In thi limit <n > reduce to <n >= e,(",) (30) Plugging thi into (26), we get <n >= e,(",) = e e," = N (3) N e = P e," (32) Thu <n >= N e," P e," (33) Hence we ee that in the claical limit of uciently low denity uciently high temperature, the Fermi{Dirac and Boe{Eintein ditribution law reduce to the Maxwell{ Boltzmann ditribution. One can alo how that the claical limit crepond to the cae where the average ditance between the particle i much larger than the ize of the mean de Broglie wavelength <>aociated with each particle h <>=2 <p> where < p > i the mean momentum of a particle. ociating a wavelength with a particle i part of wave{particle duality. 6 (34)

Physics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014

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