The Final Lecture (#40): ReviewThe Final Lecture (#40): ReviewChapters 1Chapters 1--10, Wednesday April 2310, Wednesday April 23rdrd• Announcements• Homework statistics• Finish review of third exam• Quiz (not necessarily in this order)• Review Chapters 3 to 7Reading: Reading: Chapters 1Chapters 1--10 (pages 1 10 (pages 1 --207)207)Final: Wed. 30th, 5:30Final: Wed. 30th, 5:30--7:30pm in here7:30pm in hereExam will be cumulativeExam will be cumulativeHomework StatisticsHomework Statistics20 40 60 80 10002468101214Mean = 81%Median = 88%Number of studentsScore (%)Review of Review of Chapters 3 & 4Chapters 3 & 4Classical and statistical probabilityClassical and statistical probabilityClassical probability:•Consider all possible outcomes (simple events) of a process (e.g. a game).•Assign an equal probability to each outcome.Let W = number of possible outcomes (ways)Assign probability pito the ithoutcome11&1iiippWWW==×=∑Classical and statistical probabilityClassical and statistical probabilityStatistical probability:•Probability determined by measurement (experiment).•Measure frequency of occurrence.•Not all outcomes necessarily have equal probability.••Make Make N N trialstrials••Suppose Suppose iiththoutcome occurs outcome occurs nniitimestimeslimiiNnpN→∞⎛⎞=⎜⎟⎝⎠012345-3.0-2.5-2.0-1.5-1.0-0.50.0Nσ10.510 0.15100 0.041000 0.013210000 0.00356100000 0.00145log(σ )log(N)()()log log0.516aNbaσ=+=−Statistical fluctuationsStatistical fluctuations1/2Nσ−∝1/2 1/2Error: , Relative error ( / )ii ii inn nn n−Δ∝ Δ ∝The axioms of probability theoryThe axioms of probability theory1. pi≥ 0, i.e. piis positive or zero2. pi≤ 1, i.e. piis less than or equal to 13. For mutually exclusive events, probabilities add, i.e.••Compound events, (Compound events, (ii+ + jj): this means either event ): this means either event iioccurs, or event occurs, or event jjoccurs, or both.occurs, or both.••Mutually exclusive: events Mutually exclusive: events iiand and jjare said to be mutually exclusive are said to be mutually exclusive if it is impossible for both outcomes (events) to occur in a sinif it is impossible for both outcomes (events) to occur in a single gle trial.trial.12........rpp p p=++ +••In general, for In general, for rrmutually exclusive events, the probability that one mutually exclusive events, the probability that one of the of the rrevents occurs is given by:events occurs is given by:Independent eventsIndependent eventsExample:What is the probability of What is the probability of rolling two sixes?rolling two sixes?Classical probabilities:Classical probabilities:166p=Two sixes:Two sixes:11 16,666 36p=×=•Truly independent events always satisfy this property.•In general, probability of occurrence of r independent events is:12........rppp p=×× ×nixiStatistical distributionsStatistical distributions879106,whereiiiiinxxNnN==∑∑Mean:Statistical distributionsStatistical distributionsnixi16,where limiii iiNnxpx pN→∞==∑Mean:N →∞Statistical distributionsStatistical distributionsnixi16() ()22iiixpxxσ=Δ = −∑Standard deviation()221() exp22xxpxσσπ⎧⎫−⎪⎪=−⎨⎬⎪⎪⎩⎭Statistical distributionsStatistical distributionsGaussian distribution(Bell curve)Statistical Mechanics Statistical Mechanics ––ideas and definitionsideas and definitionsA quantum state, or microstateA quantum state, or microstate••A unique configuration.A unique configuration.••To know that it is unique, we must specify it as To know that it is unique, we must specify it as completely as possible...completely as possible...Classical probabilityClassical probability••Cannot use statistical probability.Cannot use statistical probability.••Thus, we are forced to use classical probability.Thus, we are forced to use classical probability.An ensembleAn ensemble••A collection of separate systems prepared in A collection of separate systems prepared in precisely the same way.precisely the same way.Statistical Mechanics Statistical Mechanics ––ideas and definitionsideas and definitionsThe microcanonical ensemble:The microcanonical ensemble:Each system has same:Each system has same:# of particles# of particlesTotal energyTotal energyVolumeVolumeShapeShapeMagnetic fieldMagnetic fieldElectric fieldElectric fieldand so on....and so on................These variables (parameters) specify the These variables (parameters) specify the ‘‘macrostatemacrostate’’of the ensemble. A macrostate is specified by of the ensemble. A macrostate is specified by ‘‘an an equation of stateequation of state’’. Many, many different microstates . Many, many different microstates might correspond to the same macrostate.might correspond to the same macrostate.Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×Ensembles and quantum states (microstates)Ensembles and quantum states (microstates)Cell volume, Cell volume, ΔΔVVVolume Volume VV10 particles, 36 cells10 particles, 36 cells1016136310ip−⎛⎞=⎜⎟⎝⎠≈×EntropyEntropyBoltzmann hypothesis: the entropy of a system is related to the probability of its being in a state.() ()1npSfWWWφ=⇒==()lnBSk W=Rubber band modelRubber band modeld()()!!,!! ! !NNWNnnn n N n++− + +==−()()ln ln ln lnWNNn n Nn Nn+++ +=−−− −SterlingSterling’’s approximation: s approximation: ln(ln(NN!) = !) = NNlnlnNN−−NN1111ln ln2222xxxxN++−−⎧⎫⎛⎞⎛⎞⎛⎞⎛⎞=− +⎨⎬⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠⎩⎭Chapters 5Chapters 5--77•Canonical ensemble and Boltzmann probability•The bridge to thermodynamics through Z•Equipartition of energy & example quantum systems•Identical particles and quantum statistics•Spin and symmetry•Density of states•The Maxwell distributionReview of main results from lecture 15Review of main results from lecture 15Canonical ensemble leads to Boltzmann distribution function:()()()exp / exp /exp /iB iBijBjEkT EkTpZEkT−−==−∑Partition function:()exp /jjBjZgEkT=−∑Degeneracy: gjEntropy in the Canonical EnsembleEntropy in the Canonical EnsembleM systemsniin state ψi12!! !.. !..MiMWnn n=ln lniiMBBiiiinnSkM kMppMM⎛⎞⎛⎞=− =−⎜⎟⎜⎟⎝⎠⎝⎠∑∑lnBiiiSk pp=−∑Entropy per system:The bridge to thermodynamics through