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UF PHY 4523 - Canonical ensemble and Boltzmann probability

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Lecture 26 — Review for Exam II Lecture 26 — Review for Exam II Chapters 5-7, Chapters 5-7, 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 - 159)All of chapters 5 to 7 (pages 91 - 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 /i B i Bij BjE k T E k TpZE k T- -= =-�Partition function:( )exp /j j BjZ g E k T= -�Degeneracy: gjEntropy in the Canonical EnsembleEntropy in the Canonical EnsembleM systemsni in state i1 2!! !.. !..MiMWn n n=ln lni iM B B i ii in nS k M k M p pM M� � � �=- =-� � � �� � � �� �lnB i iiS k p p=-�Entropy per system:The bridge to thermodynamics The bridge to thermodynamics through through ZZ( )exp / ;j BjZ E k T= -�js represent different configurationslnBF k T Z=-( ) ( )ln lnlnB BVV VT Z ZFS k k Z TT T T� �� �� � � ��� �� �=- =- = +� �� �� � � �� � �� �� �� � � ��( ) ( )2 2ln lnln lnB B BV VZ ZU TS F k T Z T k T Z k TT T� �� �� � � �� �= + = + - =� �� � � �� �� �� � � ��22VV VVU S FC T TT T T� �� � �� � � �= = =-� � � �� �� � �� � � �� �A single particle in a one-dimensional A single particle in a one-dimensional boxboxV(x)V = ∞ V = 0V = ∞xx = Lsinnn xALpf� �=� �� �0f =0f =2 2 2222nnnmLpe g� �= =� �� �hThe three-dimensional, time-independent SchrThe three-dimensional, time-independent Schrödinger equation:ödinger equation:( ) ( ) ( ) ( )22, , , , , , , ,2x y z V x y z x y z x y zmff ef- � + =h2 2 222 2 2x y z� � �� = + +� � �A single particle in a three-dimensional A single particle in a three-dimensional boxbox( )1 2 31 2 3, , sin sin sin , ,in x n y n zx y z A n n nL L Lp p pf� � � � � �= =� � � � � �� � � � � �( )2 22 2 21 2 32, 1,2,3...2n in n n nmLpe� �= + + =� �� �hFactorizing the partition functionFactorizing the partition function22 231 21 2 322 231 21 2 322 231 2trans1 1 11 1 11 2 30 0 03/3/ 223 3/ 22 3 22 2nn nn n nnn nn n nnn nB BDZ e e ee e ee dn e dn e dnmk T V mk LL Tgg ggg ggg gp l p� � �-- -= = =� � �-- -= = =� � �-- -=� �� �� �=� �� �� �� �� �� �� �� �� �=� �� �� �� �� �� �� �� �= = =� �� �� �� ����� � �� � �h h22 222mLpg =hEquipartition theoremEquipartition theorem22 231 21 2 3trans 1 2 31 1 1nn nn n nZ e e e Z Z Zgg g� � �-- -= = == = � ����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.( ) ( )1 2 3 1 2 3ln lnNN NZ Z Z Z Z N Z Z Z= � � � =( )1 2 3ln ln lnBF Nk 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 1/2kB to the heat capacity.( ) ( )1 2 3 1 2 3ln ln ln lnB BF k T Z Z Z k T Z Z Z=- =- + +Also,Rotational energy levels for diatomic Rotational energy levels for diatomic moleculesmolecules( )2122 1lll lIg le = += +hI = momentof inertial = 0, 1, 2... is angular momentum quantum numberCO2I2HI HCl H2R(K) 0.56 0.053 9.4 15.3 88Vibrational energy levels for diatomic Vibrational energy levels for diatomic moleculesmolecules( )12nne w= + h = naturalfrequency ofvibrationn = 0, 1, 2... (harmonic quantum number)I2F2HCl H2V(K) 309 1280 4300 6330Specific heat at constant pressure for HSpecific heat at constant pressure for H22CP (J.mol1.K1)52R72R92RHH22 boils boilsTranslationTranslationCCPP = = CCVV + + nRnRExamples of degrees of freedom:Examples of degrees of freedom:( )( )2 21 12 22 21 12 22 2 2322 21 1,2 21 12 21 12 21212average, or r.m.s. valueLC B BHO B Btrans x y z Brot dia x y B BE C V L i k T k TE k x m v k T k TE m v v v k TE I k T k Tw w= + = += + = += + + == + = +�BosonsBosons( ) ( ) ( ) ( ) ( ){ }( )2,Bose 1 2 1 2 2 1 2,Bose 2 11, ,2i j i jx x x x x x x xy ff ff y= + =( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )3,Bose 1 2 2 1 2 3 2 1 32 3 1 3 2 11 1 2 1 3 2, ,i j k i j ki j k i j ki j k i j kx x x x x x x x xx x x x x xx x x x x xy ff ff ffff ff ffff ff ff= ++ ++ +•Wavefunction symmetric with respect to exchange. There are N! terms.•Another way to describe an N particle system:1 2 31 1 2 2 3 3, , ,iin n nE n n nye e e= ����= + + +����•The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wavefunction i.•For bosons, occupation numbers can be zero or ANY positive integer.FermionsFermions( ) ( ) ( ) ( ) ( ){ }( )2,Fermi 1 2 1 2 2 1 2,Fermi 2 11, ,2i j i jx x x x x x x xy ff ff y= - =-•Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.:( )1 1 13,Fermi 1 2 3 2 2 23 3 3( ) ( ) ( ), , ( ) ( ) ( )( ) ( ) ( )i j ki j ki j kx x xx x x x …


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