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Example: Roll two dice (N = 2, q = 6)1(6,6)1236 microstates11 macrostates4(3,6) (4,5) (5,4) (6,3)95(2,6) (3,5) (4,4) (5,3) (6,2)86(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)74(1,4) (2,3) (3,2) (4,1)53(1,3) (2,2) (3,1)42(1,2) (2,1)31(1,1)23(4,6) (5,5) (6,4)105(1,5) (2,4) (3,3) (4,2) (5,1)62(5,6) (6,5)11MultiplicityAccessible microstatesMacrostate (sum)States of model systemsEach macrostate is associated with energy E, but there may be many microstates with the same energy!D: Multiplicity of a macrostate = # of microstates Note: Each state has to be counted separatelyExample: quantum states and energy levels of atomic systems: 1. Hydrogen (1 e + 1 p) 2. Lithium (3 e + 3 p + 3(4) n)3. Boron (5 e + 5 p + 5(6) n)Example: one spinless particle in a boxSchrödinger equation:Wavefunction (standing waves):Energy spectrum:Macrostates, microstates, and multiplicites:)(2),,(2222zyxzyxnnnLMnnn ++⎟⎠⎞⎜⎝⎛=πεh......6)3,1,2(),3,2,1(),1,3,2(),2,3,1(),2,1,3(),1,2,3(1)2,2,2(3)3,1,1(),1,3,1(),1,1,3(3)2,2,1(),2,1,2(),1,2,2(3)2,1,1(),1,2,1(),1,1,2(1)1,1,1()/sin()/sin()/sin(),,( LxnLxnLxnAzyxzyxπππ=Ψ),,(),,(222zyxzyxMΨ⋅=Ψ∇−εh2222222zyx ∂∂+∂∂+∂∂≡∇Binary Model SystemIn general, a model system has N sites (particles) and each site can be in q states with equal probability (1/q)total # of microstates: qNIf q = 2, e.g., each site is capable of only two states (+/-, up/down, A/B), we have binary model systemtotal # of microstates for a binary system: 2NExample: A system of N=3 elementary magnets (spins) with moment mMicrostate is given by defining an orientation on each siteOne possible microstate:Order is important!Macrostate can be related to a total magnetic moment (the sum)Corresponding macrostate: M = m –m –m = –mOrder is not important!Example: N=3 binary magnets1(-m, -m, -m)-3 m[none up]8 microstates4 macrostates3(m, m, -m) (m, -m, m) (-m, m, m)1 m[two up]1(m, m, m)3 m[all three up]3(-m, -m, m) (-m, m, -m) (m, -m, -m)-1 m[one up]MultiplicityAccessiblemicrostatesMacrostateMicrostates and macrostates of N magnetsFor large N, 2N >> N + 1and there many more microstates that observable macrostates!Example: N = 10, 210= 1024 >> 10+1=11. Q: Why the # of macrostates is smaller then the # of microstates?A: (Mostly) because macrostate does not distinguish between sites! It only cares about total number of states with spins up and total number of states with spin down. N+1Nm, (N-2)m,…, -Nm2N(m, m,…, m ), (-m, m,…, m ), …,(-m, -m,…, -m )N……………43m, m, -m, -3m8(m, m, m),(-m, m, m),..., (-m, -m, -m)332m, 0, -2m4(m, m),(m, -m),(-m, m),(-m, -m)22m, -m2(m),(-m)1#Macrostates (M)#Accessible Microstates#Microstates of N binary magnetsNotation:Microstate is associated with a particular configuration, e.g.,Generating function: all microstates are contained in a symbolic product which has 2N termsNumber of microstates: 2NExample: N = 2Example: N = 3()()()() NN↓+↑⋅⋅⋅↓+↑↓+↑↓+↑332211 N↓⋅⋅⋅↓↑↑↑↓↑↓7654321 ()mm −↓≡+↑≡ ; ()()212121212211 ↓↓+↑↓+↓↑+↑↑=↓+↑↓+↑ ()()().... 321321321332211+↑↓↑+↓↑↑+↑↑↑=↓+↑↓+↑↓+↑Macrostates of N binary magnetsMacrostate can be associated with total magnetic moment M(directly proportional to the energy in applied magnetic field)M varies from –Nm to +mN: M = Nm, (N-2)m, (N-4)m, …., -Nm: N+1 possible macrostates(obtained from fully polarized state by flipping magnets one at a time…) Number of macrostates: N+1Example: 1). M=Nm2). M=(N-2)mone spin is flipped where,1mmmMiNii±==∑=44444434444442148476smicrostate identicalcally macroscopi spins ..............NN↓↑↑↑↑↑↓↑↑↓↑↑4342148476microstate 1spins ...N↑↑↑↑Enumeration of StatesNumber of spins, N:Spin excess, 2s (total spin):Note: Spin excess fully defines the macrostateSince macrostate doesn’t care about particular sites, we can drop site labels in the generating function:Example: N = 2()( )()()()NNN↓+↑⎯→⎯↓+↑⋅⋅⋅↓+↑↓+↑↓+↑ 332211 sNNsNN−=↓+=↑↓↑21 :states spin- ofNumber 21 :states spin- ofNumber sNN 2=−↓↑NNN=+↓↑()()()↓↓+↑↓+=↑↑↓+↑⎯→⎯↓↓+↑↓+↓↑+↑↑=↓+↑↓+↑2 2212121212211Binomial Series (reference)http://mathworld.wolfram.com/BinomialSeries.htmlBinomial Coefficient (reference)http://mathworld.wolfram.com/BinomialCoefficient.htmlMultiplicity functionTo find we can use binomial expansion:Change variables: ()tNtNtNYXttNNYX−=∑−=+0!)!(! ()N↓+↑()()NNNNsNtYXNsNtNNsN t↓+↑→+⎯→⎯=−=−=+=∑∑−==↓↑ ;21 ;2121210sNsNNNsNsNgsNsNN−+−=↓↑⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛+=↓+↑∑21212121),(!21!21!)( 444344421- Multiplicity functionMultiplicity functionMultiplicity function: # of microstates with the same total spin s.In magnetic field s is directly related to energy, g is the multiplicity (degeneracy) of an energy level having spin excess sCheck total # of microstates:adding one spin doubles the # of microstatesAnother application is binary alloy system…()⎟⎟⎠⎞⎜⎜⎝⎛−≡⎟⎟⎠⎞⎜⎜⎝⎛+≡=⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛+=↓↑sNNsNNNNNsNsNNsNg2121!!!!21!21!, ()sNsNNNsNsNg−+−=↓↑=↓+↑∑21212121),( ()NNNNssNg 211),( 2121=+=∑−=Multiplicity functionStability of the thermodynamic systems follows from the sharpness of the peak (at s = 0) in multiplicity function:It is convenient to work with the log(g(N,s)):Stirling approximation()!21!21!, ⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛+=sNsNNsNg!log!log!log),(log ↓↑−−=NNNsNg()[]...121exp2! ++−≅ NNNNNNπ)1( >>NMultiplicity functionStirling approximation in the log form:Similarly:NNNN −⎟⎠⎞⎜⎝⎛++≅ log212log21!logπ↓↓↓↓↑↑↑↑−⎟⎠⎞⎜⎝⎛++≅−⎟⎠⎞⎜⎝⎛++≅NNNNNNNNlog212log21!loglog212log21!logππNNNNNN log21log21log212log21!log −+−⎟⎠⎞⎜⎝⎛++≅π()NNNNN −++≅ log12log21!logπMultiplicity functionSubtract:Approximate: ()NNNNNNNsNg↓↓↑↑⎟⎠⎞⎜⎝⎛+−⎟⎠⎞⎜⎝⎛+−≅ log21log2121log21,logπ()!log!log!log,log ↓↑−−=NNNsNg⇒+=↓↑NNN⎟⎠⎞⎜⎝⎛+=+=↑NsNsNN2121212222log21log2loglog⎟⎠⎞⎜⎝⎛−+−≅⎟⎠⎞⎜⎝⎛++−=↑NsNsNsNN2222log21log2loglog⎟⎠⎞⎜⎝⎛−−−≅⎟⎠⎞⎜⎝⎛−+−=↓NsNsNsNN()2211log1 xxxx −≅+⇒<<Multiplicity


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U of M PHYS 4201 - LECTURE NOTES

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