2 States of a System. 2.1 States / Configurations 2.2 Probabilities of States. 2.3 Counting States 2.4 Entropy of an ideal gas

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1 2 State of a Sytem Motly chap 1 and 2 of Kittel &Kroemer 2.1 State / Configuration 2.2 Probabilitie of State Fundamental aumption Entropy 2.3 Counting State 2.4 Entropy of an ideal ga Phyic 112 (S2012) 02 State-Entropy 1

2 Sytem State, Configuration Microcopic: each degree of freedom Claical q i, p i = q Quantum tate # i Statitical Mechanic Moment f (q i, p i ) < q i >, < p i > Thermodynamic Macrocopic Variable State Configuration State = Quantum State Well defined, unique Dicrete claical thermodynamic where entropy wa depending on reolution ΔE Configuration = Macrocopic pecification of ytem U = Macrocopic Variable Extenive: U,S,F,H,V,N Intenive: T,P,µ n i=1 U i T = 2 3k B 1 N n i=1 U i Not unique: depend of level of detail needed Variable+ contraint=> not independent Phyic 112 (S2012) 02 State-Entropy 2

3 Quantum Mechanic in 1 tranparency Fundamental potulate State of one particle i characterized by a wave function Probability ditribution = ψ x ( ) 2 with ψ ψ ψ ( x) Phyical quantity hermitian operator. => A finite ytem ha dicrete eigenvalue ψ ( x)dx = 1 In general, not fixed outcome! Expected value of O = ψ O ψ ψ ( x) Oψ ( x)dx Eigentate tate with a fixed outcome e.g., O ψ >= o ψ > where o i a number. i t = E i x j ϕ(x,t) = = p j 1 2π 2 "State" of well defined momentum 3/2 e i Et p x E = p2 2M => 2 2M 2 Ψ = i t Ψ Eigenvalue of energy ε: 2 2M 2 Ψ = εψ Phyic 112 (S2012) 02 State-Entropy 3

4 Graphic Example ψ ψ dimenional cae, infinitely deep quare well ψ = Ain n exp i n xπ x xπ x exp i n xπ x = A 2i 2 Solution of the Schrödinger equation 2M 2 Ψ 2 2M with the proper boundary condition ε = 2 n x π 2M 0 ψ 2 dx = 1 A = 2 2 ψ x 2 = εψ i x ψ = p ψ uperpoition of 2 State of well defined momentum p = ± n xπ 2 ε = p2 2M Phyic 112 (S2012) 02 State-Entropy 4

5 Prototype: n=2 n=1 Ideal ga of particle Free particle propagating in pace: orbital ψ(x) ( motion part of the wave function) Single particle in a box : Cubic ide cf. K&K p.72 Ψ (x, y,z) = Ain n x πx in n y πy in n z πz Phyic 112 (S2012) 02 State-Entropy Quantum State Atomic level cf Fig 1.1 Kittel and Krömer Have to take into account multiplicity if we conider degenerate tate (i.e. have the ame energy) Spin in general 2+1 tate (exception photon =1 but 2 tate) Spin 1/2 =±1/2 If magnetic field oriented along +/- direction, the energy = -/+mb (m=magnetic moment) Note: Idea of an iolated ytem of pin i omewhat trange <= tranition between higher and lower energy tate. Iolated = electromagnetic emiion reaborbed or reflected back n x,n y,n z integer > 0 ε nx,n y,n z = 2 π 2 n 2M 2 x + n 2 2 y + n z Ideal ( non interacting): orbital not ditorted by preence of N particle. Wave function = product of ingle wave function. Can bounce againt each other provided interaction are hort! 5 ( )

6 Fundamental Potulate Probabilitic decription of tate of a ytem Becaue of microcopic procee, the ytem experience light fluctuation: decribed by probability of being in tate i. Equilibrium = No net Flux =>Stationary No evolution of probability ditribution with time Iolated = cloed : No energy/particle exchange with outide volume contraint OK An iolated ytem in equilibrium i equally likely to be in any of it acceible tate Note: Kittel doe not pecify "in equilibrium" doe not matter in cae where we are cloe to equilibrium "Acceible" e.g. Conervation of energy +Some tate may not be acceible within time cale of experiment Phyic 112 (S2012) 02 State-Entropy 6

7 Conequence Probability of a configuration (iolated, in equilibrium) If we call g the number of tate for a given macrocopic pecification of the configuration, and g t the total number of tate acceible to the ytem Number of tate in configuration Prob(configuration)= = g Total number of tate g t Thi allow u to compute probabilitie of configuration! Thi computation method i conventionally called a microcanonical computation Phyic 112 (S2012) 02 State-Entropy

8 Remark Probabilitic decription of tate of a ytem Two poible definition: Obervation of different identical ytem at a given time "enemble" of ytem j Obervation at different time, ytem wander over it acceible tate Ergodicity: under very general condition, we obtain identical reult T < y > enemble =< y > time lim 1 T T o y(t)dt The fundamental potulate can be demontrated in Quantum Mechanic Celebrated H theorem Phyic 112 (S2012) 02 State-Entropy 8

9 Entropy Definition Entropy Kittel (our i more general) σ = p tate log p = H H="Negentropy" = "Information" of Shannon σ = log( g t ) For an iolated ytem in equilibrium: identical to Kittel p = 1 + equiprobable ( ) p = 1 g t σ = log g t where g t i the total number of acceible quantum tate in the configuration Note: In claical thermodynamic, definition uually ued dq dq dq S = k B σ = σ = T = k B T with τ = k τ B T k B = J / K where k B i the Boltzmann contant (ue implicitly 3rd law of thermodynamic S=0 at T=0) Entropy require the ue of quantum mechanic: it involve the Planck contant Phyic 112 (S2012) 02 State-Entropy 9

10 H theorem: an intuitive look Theorem: An iolated ytem evolve toward a configuration where all the tate are equiprobable and the entropy i maximum Baed on the fact that probability of tranition per unit time from r to i equal to probability of tranition per unit time from to r Γ r = Γ r Suppoe we have initially N r particle in r and N <<N r in r, the number of particle going from r to in time Δt i much bigger than from to r. N r Γ r Δt >> N Γ r Δt Therefore N r decreae and N increae, till they are in average equal. The probabilitie of and r are then equal! N r Γ r Δt N Γ r Δt Phyic 112 (S2012) 02 State-Entropy r 10

11 Optional H theorem (Optional) Modern Verion: ee Reif Appendix A12 Conider an iolated ytem, and quantum tate. The probabilitie of occupancy p are uch that p = 1 Quantum mechanic: Probabilitie per unit time of tranition between tate r and are ymmetric Γ r = Γ r W r 2 Starting from the detailed balance argument Conider now We have or equally well Add the two quantitie ( ) H p log p = σ dh dp = r dt r dt dh = Γ dt r ( p r p ) r dh dt dp r dt ( log( p r ) +1) = Γ r p p r = 1 2 r = p Γ r p r Γ r = Γ r p p r ( ) ( log( p r ) +1) ( log( p ) +1) = Γ r p p r Γ r ( ) p p r If p > p r, log( p r ) > log( p ) and vice vera + Γ r > 0 r r ( ) ( log( p ) log( p r )) dh dt 0 dσ dt ( ) ( log( p ) +1) 0 Equality hold if p = p r : => at equilibrium quantum tate have equal probabilitie! Phyic 112 (S2012) 02 State-Entropy 11

12 Remark Thi proof i in Reif Related, a Boltzmann already knew, to the ymmetry of tranition probabilitie Probability of tranition per unit time tate r tate = Prob of tranition tate tate r Γ r = Γ r Quantum Mechanic -> Statitical mechanic if tate looe their coherence (generally true) =>Quantum mechanic i compatible with tatitical mechanic! Simulation (5 State ): S2009/Phy112/Htheorem/Htheorem.html Diffuion in pace tate Phyic 112 (S2012) 02 State-Entropy 12

13 Counting State: Dicrete State Preliminarie Number of permutation between N object = factorial N!= N(N 1)(N 2) = Γ (N +1) Stirling approximation log N! N Independent pin 1/2 g() = N log N N N pin, each of them ha two tate (up, down) If we define a configuration by the number of pin up, the number of tate in the configuration number of way to chooe n up object g(n up,n down ) = N ( N 1).. ( N n + 1 up ) N! = with n down = N n up n up! n up!n down! Total pin = n up n down We could have intead choen to label the configuration by the difference (proportional to total total pin): uppoe N i even N! N 2 +! N 2! Phyic 112 (S2012) 02 State-Entropy 13 log ( 2πN ) ( ) 1 2 n up = N / 2 + n down = N / 2 N 2N 1 We do not care about order! 2πN / 4 exp N / 4 Gauian! (from Sterling approximation)

14 Counting State: Particle Denity of patial tate per unit phae pace Phae pace element for a ingle particle in 3 dimenion: Theorem: the denity of patial tate (orbital) per unit phae pace for a ingle particle in 3 dimenion i 1/h 3 = denity of quantum tate for a pinle particle Proof: Not in book! Conider a particle in a box. It patial wave function i Ψ(x,y,z) = Ain n πx x in n πy y in n πz z with n x,n y,n z integer > 0 What i the x component of the momentum? It i not well defined! d 3 x d 3 p in n πx x = ei n x πx e i n xπx 2i Applying p = i one ee thi i the uperpoition of 2 momenta n x π and - n xπ Intuitively the particle i going back and forth. Thi i a direct conequence of the Heienberg Uncertainty Principle Contraining the particle in pace, you cannot have a well defined momentum! Phyic 112 (S2012) 02 State-Entropy 14

15 Phyic 112 (S2012) 02 State-Entropy Proof et u work firt in one dimenion: Conider the x direction, For all tate, the poition coordinate range over et u tart with n x =1. The x momentum pan Δp x = π π = 2 π poitive p x negative p x The x component of the phae pace volume panned by the n x =1 tate i therefore Increaing n x by 1 will add one more tate and increae the momentum pace panned by the phae pace volume by an additional ΔxΔp x = 2 π = h and o on. Therefore the number of tate per unit phae pace i h. Thi i a totally general and exact reult: the phae pace denity i 1/h for each of the dimenion! In 3 dimenion the phae pace volume occupied by each of the tate n x = 1,n y = 1,n z = 1 n x = 2,n y = 1,n z = 1 n x = 1,n y = 2,n z = 1 n x = 1,n y = 1,n z = 3... ΔxΔp x = 2 π = h i h 3 => the phae pace denity of tate i 1/h 3 15 Δx = x 2 π p x

16 Homework Counting State: Particle Proof à la Kittel Not explicitly in book but many uch type of calculation through out Conider again a particle in a cubic box, and compute number of tate between E and E+dE = number of integer n x,n y,n z uch that E 2 π 2 n 2M 2 x + n 2 2 ( y + n z ) < E + de In the n x,n y,n z pace, each tate correpond to a volume of unity For n x,n y,n z large enough, the number of tate i given by the volume of one quadrant of pherical hell of radiu n = n 2 x + n 2 y + n 2 z = 2ME 2M and thickne dn = E π 2π de n Continuou # of tate = 1 ( 8 4πn2 dn = 2π 3 2M ) 3/2 E n approximation y de 8π 3 3 quadrant Volume of phae pace element uch that the energy i between E and E+dE i 3 4πp 2 dp with E = p2 i π 23 2M 2M # State Phae pace volume = 1 8π 3 = 1 3 h 3 ( ) 3/2 EdE n x n z Phyic 112 (S2012) 02 State-Entropy 16

17 Now conider N particle in weak interaction Calculation of number of tate a a function of U N particle Weak interaction => # tate = with the contraint that total energy i U = U where M i the ma part i j =1 2M Space integral d 3 x i = V N i Momentum integral: we need to conerve energy 1 particle in 1 dimenion: only 1 momentum tate with right energy in 2 dimenion circumference of circle in 3 dimenion Ideal Ga g = i d 3 x i d 3 p i h 3 p 2 + p2 2 = 2MU δ p1 2 + p MU d 2 p = 2π 2MU i area of 2-phere p 2 + p2 2 + p3 2 = 2MU δ p1 2 + p2 2 + p MU d 3 p = 4π 2MU i N particle with 3 momentum component area of (3N-1)-phere <= 3 xn dimenion in total N 3 2 δ p ij 2MU part i j =1 for large N g V N U N 3N 2 part 3 p ij 2 d 3 p i U 3N 1 2 ( ) 2 Phyic 112 (S2012) 02 State-Entropy 17

18 Ideal Ga law Why i it important? let u aume that the entropy we have defined i the entropy of thermodynamic σ = p i log p i = log g t i Define τ = k b T ( ) = S k b The Thermodynamic identity Firt law du = TdS pdv quantity of heat work by preure tranfered to ytem on the ytem τdσ pdv τ = U From previou lide σ = log V N U 3/2 N σ V,N ( ) + cont. U = Aexp τ = U σ V,N = 2 3N U U = 3 2 Nτ 3 2 Nk bt ( ) i then (no change of number of particle N) p = U V σ,n 2 3N σ V 2/3 (A i a contant) indeed look like the temperature p = U = 2 U V σ,n 3 V pv = Nτ Nk bt indeed look like the preure Phyic 112 (S2012) 02 State-Entropy 18

19 Our and Kittel approach We will define in chapter 3 1 σ U,V, N = ( ) U ( ) τ p σ U,V, N = τ U dσ 1 τ du + p dv du τdσ pdv τ and how that thee definition behave a the temperature and preure Phyic 112 (S2012) 02 State-Entropy 19

20 Optional Ideal Ga 2 technical note (optional) 1) Exact formula: It can be hown that the urface area of 3N-1 phere of radiu r i: r 3N 1 Ω 3N where the olid angle factor i ( )3N 2 Ω 3N = 2 π Γ 3N 2 3N = 2(π ) 2 ( 3 2 N 1)! with m 2! = m 2 m 2! = m 2 m ( ) m even m π 1/2 m odd See e.g. Wolfram ite: math/h/h456.htm or 2) If you follow carefully the dimenion, we need a factor h/ g = h δ N 3 p 2 2MU d 3 x d 3 p i i ij part i j =1 h 3 = V N 1/3 2MU 3N 1 i h δ function have a dimenion! δ ( x) dx = δ f x We can compute (painfully) the number of tate and then get probabilitie: Microcanonical method 3N 1 ( ) 2 Ω 3N ( ) df dx dx = 1 Phyic 112 (S2012) 02 State-Entropy 20

21 Homework Sackur Tetrode Formula We are now in poition to compute the entropy we could try σ = log g = log V N 1/3 3N 1 ( 2MU ) 2 Ω 3N 3N 1 h Only problem: violently diagree with experiment! Solution: In quantum mechanic, particle are inditinguihable We have over-counted the number of particle by N! (Gibb) Sackur Tetrode Monoatomic, pinle: cf Kittel chapter 6 3N 1 ( ) 2 Ω 3N g = 1 h N! δ N 3 p 2 2MU d 3 x d 3 p i i ij part i j =1 h 3 = V N 1/3 2MU 3N 1 i N!h For large N we can ue the Stirling approximation both for integer and half integer log m 2! m 2 log m 2 m 2 and we get 3/2 2π M 2U σ(u,v, N) = log g N log N h 2 3N + N log V N + 5 / 2 Writing n = N V,n Q = Phyic 112 (S2012) 02 State-Entropy 2π M h 2 2U 3N 3/2 21 we get σ = S = N log n Q k B n + 5 2