Lecture 26 Lecture 26 ——Review for Exam II Review for Exam II Chapters 5Chapters 5--7, 7, Monday March 17Monday March 17thth•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 distributionReading: Reading: All of chapters 5 to 7 (pages 91 All of chapters 5 to 7 (pages 91 --159)159)Exam 2 on Wednesday, in classExam 2 on Wednesday, in classNext homework due next weekNext homework due next weekReview 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 The bridge to thermodynamics through ZZ()exp / ;jBjZEkT=−∑jsrepresent different configurationslnBFkTZ=−()()ln lnlnBBVVVTZ ZFSk kZTTT T⎧⎫∂∂⎡⎤⎡⎤∂⎪⎪⎛⎞=− =− = +⎨⎬⎜⎟⎢⎥⎢⎥∂∂ ∂⎝⎠⎪⎪⎣⎦⎣⎦⎩⎭()()22ln lnln lnBBBVVZZUTSFkTZT kTZkTTT⎧⎫∂∂⎡⎤⎡⎤⎪⎪=+= + − =⎨⎬⎢⎥⎢⎥∂∂⎪⎪⎣⎦⎣⎦⎩⎭22VVVVUS FCTTTT T⎛⎞∂∂ ∂⎛⎞ ⎛⎞== =−⎜⎟ ⎜⎟⎜⎟∂∂ ∂⎝⎠ ⎝⎠⎝⎠A single particle in a oneA single particle in a one--dimensional boxdimensional boxV(x)V = ∞ V = 0V = ∞xx = LsinnnxALπφ⎛⎞=⎜⎟⎝⎠0φ=0φ=222222nnnmLπεγ⎛⎞==⎜⎟⎝⎠=The threeThe three--dimensional, timedimensional, time--independent Schrindependent Schröödinger equation:dinger equation:()()()()22,, ,, ,, ,,2xyz V xyz xyz xyzmφφεφ−∇ + ==2222222xyz∂∂∂∇= + +∂∂∂A single particle in a threeA single particle in a three--dimensional boxdimensional box()12 3123, , sin sin sin , ,inx ny nzxyzA nnnLLLπππφ⎛⎞⎛ ⎞⎛⎞==⎜⎟⎜ ⎟⎜⎟⎝⎠⎝ ⎠⎝⎠()2222 212 32, 1,2,3...2ninn n nmLπε⎛⎞=++=⎜⎟⎝⎠=Factorizing the partition functionFactorizing the partition function222312123222312123222312trans11111112300 03/3/2233/223 222nnnnnnnnnnnnnnnBBDZeeeeeeedn edn ednmk T V mk LLTγγγγγγγγγπλ π∞∞∞−−−===∞∞∞−−−===∞∞ ∞−−−=⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎛⎞⎛⎞===⎜⎟⎜⎟⎝⎠⎝⎠∑∑∑∑∑∑∫∫∫==22222mLπγ==Equipartition theoremEquipartition theorem222312123trans 1 2 3111nnnnnnZeee ZZZγγγ∞∞∞−−−=====××∑∑∑If the energy can be written as a sum of independent terms, then the partition function can be written as a product of the partition functions due to each contribution to the energy.() ()123 123ln lnNNNZZZZ Z N ZZZ=×× ⇒ =()123ln ln lnBFNk T Z Z Z=− + +free energy may be written as a sum. It is in this way that each degree of freedom ends up contributing 1/2kBto the heat capacity.() ( )123 1 2 3ln ln ln lnBBFkT ZZZ kT Z Z Z=− =− + +Also,Rotational energy levels for diatomic moleculesRotational energy levels for diatomic molecules()21221llllIglε=+=+=I = momentof inertial = 0, 1, 2... is angular momentum quantum numberCO2I2HI HCl H2θR(K) 0.56 0.053 9.4 15.3 88Vibrational energy levels for diatomic moleculesVibrational energy levels for diatomic molecules()12nnεω=+=ω= naturalfrequency ofvibrationn = 0, 1, 2... (harmonic quantum number)I2F2HCl H2θV(K) 309 1280 4300 6330ωSpecific heat at constant pressure for HSpecific heat at constant pressure for H22CP(J.mol−1.K−1)ω52R72R92RHH22boilsboilsTranslationTranslationCCPP= = CCVV+ + nRnRExamples of degrees of freedom:Examples of degrees of freedom:()()221122221122222322211,22112211221212average, or r.m.s. valueLC B BHO B Btrans x y z Brot dia x y B BECV Li kT kTEk x mv kT kTE mvvv kTEIkTkTωω=+=+=+ =+=++==+=+≡BosonsBosons() ()() ()(){}()2,Bose 1 2 1 2 2 1 2,Bose 2 11,,2ij i jxxxxxxxxψφφ φφ ψ=+=()()()()()()()()()() ()()()() () () () () ()3,Bose 1 2 2 1 2 3 2 1 3231 321112 13 2,,ijk i jkijk ijkijk ijkxxx x x x x x xxxx xxxxxx xxxψφφφφφφφφφ φφφφφφ φφ φ=+++++• Wavefunction symmetric with respect to exchange. There are N! terms.• Another way to describe an N particle system:12311 2 2 33,,,iinnnEn n nψεεε=⋅⋅⋅⋅=+++⋅⋅⋅⋅• The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wavefunctionφi.• For bosons, occupation numbers can be zero or ANY positive integer.FermionsFermions() ()() () (){}()2,Fermi 1 2 1 2 2 1 2,Fermi 2 11,,2ij ijxxxxxx xxψφφ φφ ψ=−=−• Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.:()1113,Fermi 1 2 3 2 2 2333() () (),, () () ()() () ()ijkijkijkxxxxxx x x xxxxφφφψφφφφφφ=• The determinant of such a matrix has certain crucial properties:1. It changes sign if you switch any two labels, i.e. any two rows.It is antisymmetric with respect to exchange2. It is ZERO if any two columns are the same.• Thus, you cannot put two Fermions in the same single-particle state!FermionsFermions• As with bosons, there is another way to describe N particle system:12311 2 2 33,,,iinnnEn n nψεεε= ⋅⋅⋅⋅=+ + +⋅⋅⋅⋅• For Fermions, these occupation numbers can be ONLY zero or one.0ε2ε/2/3/FermiBBBkT kT kTZe e eεεε−− −=+ +BosonsBosons11 2 2 33iEn n nεεε=+ + +⋅⋅⋅⋅• For bosons, these occupation numbers can be zero or ANY positiveinteger./ 2/3/4/Bose12BBBBkT kT kT kTZeeeeεεεε−−−−=+ + + +A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):()()()()()()3121312 1423 2524//2112/2/ 2 /11 122 2//////12!jBiBBBBBBBBBBMMkTkTijkTkT kTkTkT kTkT kTkTZe eeeeeeeeeeεεεεεεεεε εεεε εεεε−−==−−−−+−+ −+−+ −+−+⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠= + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅+ + + +⋅⋅⋅⋅⋅⋅∑∑• Terms due to double occupancy – under counted.• Terms due to single occupancy – correctly counted.SO:we fixed one problem, but created another. Which is worse?•Consider the relative importance of these terms....Dense versus dilute