N96770 微奈米統計力學 1 上課地點 : 國立成功大學工程科學系越生講堂 (41X01 教室 ) N96770 微奈米統計力學

N96770 微奈米統計力學 2  Fermi-Dirac & Bose-Einstein Gases  Microcanonical Ensemble Reference: K. Huang, Statistical Mechanics, John Wiley & Sons, Inc., OUTLINES  Grand Canonical Ensemble

N96770 微奈米統計力學 3 is an eigenvector of the position operators of all particles in a system. is a vector and a state of a system. Quick Review is the wave function of the system in the state

N96770 微奈米統計力學 4 orthonormal A subset of a vector space V {v 1,…v k }, with the inner product, is called orthonormal if = 0 when i ≠ j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: |v i | = 1. : a complex number and a function of time n : a set of quantum numbers : the probability associated with n 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

N96770 微奈米統計力學 5 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. 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: electrons, protons. Examples: deuterons ( 2 H), photons.

N96770 微奈米統計力學 6 the number of states of a system having an energy eigenvalue that is between E and E+  E. N(E) : A state of an ideal system can be specified by a set of occupation numbers {n p } so that there are n p particles having the momentum p in the state. total energy total number of particles level (energy eigenvalue) n p = 0, 1, 2, … for bosons n p = 0, 1 for fermions Microcanonical Ensemble h : Planck’s constant

N96770 微奈米統計力學 7 g1g1 g2g2 g3g3 g4g4 cell Each group is called a cell and has an average energy  i. The spectrum can be divided into groups of levels containing g 1, g 2, g 3, g 4,… subcells. The levels  p become continuous as the system volume V→ ∞. The occupation number n i is the sum of n p over all levels in the i-th cell. W{n i } is the number of states corresponding to the set of occupation number {n i }.

N96770 微奈米統計力學 8 The number of ways in which n i particles can be assigned to the i-th cell. w i : For Fermions The number of particles in each of the g i subcell of the i-th cell is either 0 or 1.

N96770 微奈米統計力學 9 For Bosons Each of the g i subcell of the i-th cell can be occupied by any number of particles. Entropy : It can be shown that : the set of occupation numbers that maximizes

N96770 微奈米統計力學 10 (for bosons) (for fermions) where  : chemical potential It can be shown that (by using Stirling’s approximation) (for bosons) (for fermions) k B : Boltzmann’s constant

N96770 微奈米統計力學 11 Grand Canonical Ensemble Partition function for ideal gases where the occupation numbers {n p } are subject to the condition : the number of states corresponding to {n p } is for bosons and fermions

N96770 微奈米統計力學 12 Consider the grand partition function Z, n = 0, 1, 2, … for bosons n = 0, 1 for fermions

N96770 微奈米統計力學 13 (for bosons) (for fermions) Equations of state : (for bosons) (for fermions)

N96770 微奈米統計力學 14 Now let V → ∞, then the possible values of p become continuous. Equations of state for ideal Fermi-Dirac gases Equations of state for ideal Bose-Einstein gases

N96770 微奈米統計力學 15 Letand Then equations of state for ideal Fermi-Dirac gases become where

N96770 微奈米統計力學 16 And equations of state for ideal Bose-Einstein gases become where