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N96770 微奈米統計力學 上課地點 : 國立成功大學工程科學系越生講堂 (41X01教室) 2002.11.29 N96770 微奈米統計力學 1 OUTLINES Fermi-Dirac & Bose-Einstein Gases Microcanonical Ensemble Grand Canonical Ensemble Reference: K. Huang, Statistical Mechanics, John Wiley & Sons, Inc., 1987. 2002.11.29 N96770 微奈米統計力學 2 Quick Review | is a vector and a state of a system. | q is an eigenvector of the position operators of all particles in a system. q | (q) is the wave function of the system in the state | . 2002.11.29 N96770 微奈米統計力學 3 At any instant of time the wave function of a truly isolated system can be expressed as a complete orthonormal set of stationary wave functions { n } cn n n cn : a complex number and a function of time n : a set of quantum numbers | cn |2 : the probability associated with n orthonormal A subset of a vector space V {v1,…vk}, with the inner product <,>, is called orthonormal if <vi,vj> = 0 when i ≠ j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: |vi| = 1 . 2002.11.29 N96770 微奈米統計力學 4 Ideal Gases Two types of a system composed of N identical particles: Fermi-Dirac system The wave functions are antisymmetric under an interchange of any pair of particle coordinates. Particles with such characteristics are called fermions. Examples: electrons, protons. Bose-Einstein system The wave functions are symmetric under an interchange of any pair of particle coordinates. Particles with such characteristics are called bosons. Examples: deuterons (2H), photons. 2002.11.29 N96770 微奈米統計力學 5 Microcanonical Ensemble N(E) : the number of states of a system having an energy eigenvalue that is between E and E+E. A state of an ideal system can be specified by a set of occupation numbers {np} so that there are np particles having the momentum p in the state. E pnp total energy N np total number of particles p p np = 0, 1, 2, … for bosons np = 0, 1 for fermions p p 2 2m level (energy eigenvalue) p nh L 2002.11.29 L V 1/ 3 h : Planck’s constant N96770 微奈米統計力學 6 The levels p become continuous as the system volume V→∞. g4 The spectrum can be divided into groups of levels containing g1, g2, g3, g4,… subcells. g3 Each group is called a cell and has an average energy i. g2 The occupation number ni is the sum of np over all levels in the i-th cell. W{ni} is the number of states corresponding to the set of occupation number {ni}. 2002.11.29 N96770 微奈米統計力學 cell g1 7 N ( E ) W {ni } {ni } W {ni } wi i wi : The number of ways in which ni particles can be assigned to the i-th cell. For Fermions The number of particles in each of the gi subcell of the i-th cell is either 0 or 1. gi gi ! wi ni ni !( g i ni )! 2002.11.29 gi ! W {ni } i ni !( g i ni )! N96770 微奈米統計力學 8 For Bosons Each of the gi subcell of the i-th cell can be occupied by any number of particles. wi Entropy : (ni g i 1)! ni !( g i 1)! W {ni } i (ni g i 1)! ni !( g i 1)! S k B ln N ( E ) k B ln W {ni } {ni } It can be shown that N ( E ) W {ni } {ni }: the set of occupation numbers that maximizes W {ni } S k B ln W {ni } 2002.11.29 N96770 微奈米統計力學 9 ni ni where gi e ( i ) 1 (for bosons) 1 (for fermions) gi e ( i ) 1 (k BT ) : chemical potential kB : Boltzmann’s constant It can be shown that (by using Stirling’s approximation) ( i ) S k B g i ( i ) ln 1 e ( i ) 1 e i (for bosons) ( i ) S k B g i ( i ) ln 1 e ( i ) 1 e i (for fermions) 2002.11.29 N96770 微奈米統計力學 10 Grand Canonical Ensemble Partition function for ideal gases z (V , T ) g{n p }e E {n p } {n p } E{n p } p n p where p the occupation numbers {np} are subject to the condition : n p N p the number of states corresponding to {np} is g{n p } 1 2002.11.29 for bosons and fermions N96770 微奈米統計力學 11 Consider the grand partition function Z, Z ( , V , T ) e N z (V , T ) N 0 e N e pn p p N 0 {n p } e N 0 {n p } p e n0 ( p ) n p e ( 0 ) n0 ( 1 ) n1 n1 e ( 0 ) n0 n0 (1 ) e n1 n1 ( p ) n e p n 2002.11.29 N96770 微奈米統計力學 n = 0, 1, 2, … for bosons n = 0, 1 for fermions 12 Z ( ,V , T ) 1 1 e ( p ) Z ( , V , T ) 1 e ( p ) p (for bosons) (for fermions) p PV ln Z ( ,V , T ) k BT Equations of state : PV ( ) ln 1 e p k BT p PV ( ) ln 1 e p k BT p 2002.11.29 N96770 微奈米統計力學 (for bosons) (for fermions) 13 Now let V → ∞, then the possible values of p become continuous. dp p 0 Equations of state for ideal Fermi-Dirac gases P 4 3 k BT h N 4 3 V h p 2 ln 1 e ( p 2 2m ) dp 0 e p2 ( p 2 2 m ) 0 1 dp Equations of state for ideal Bose-Einstein gases P 4 3 k BT h p 2 ln 1 e ( p 2 2 m ) 0 1 dp ln 1 e V N 4 p2 e 3 ( p 2 2 m ) dp V h 0e V 1 e 1 2002.11.29 N96770 微奈米統計力學 14 2 2 mkBT Let e and Then equations of state for ideal Fermi-Dirac gases become P 1 3 f 5 / 2 ( ) k BT N 1 3 f 3 / 2 ( ) V where f 5 / 2 ( ) 4 x 0 2 ln 1 e x2 (1) j 1 j dx 5/ 2 j j 1 (1) j 1 j f 3 / 2 ( ) f 5 / 2 ( ) j 3/ 2 j 1 2002.11.29 N96770 微奈米統計力學 15 And equations of state for ideal Bose-Einstein gases become P 1 1 3 g 5 / 2 ( ) ln 1 k BT V N 1 3 g 3 / 2 ( ) V V 1 where g 5 / 2 ( ) 4 x 0 2 ln 1 e x2 dx j j 1 j 5/ 2 j g 3 / 2 ( ) g 5 / 2 ( ) 3 / 2 j 1 j 2002.11.29 N96770 微奈米統計力學 16