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

OUTLINES • Fermi-Dirac & Bose-Einstein Gases • Microcanonical Ensemble • Grand Canonical Ensemble Reference: K. Huang, Statistical Mechanics, John Wiley & Sons, Inc., 1987. N96770 微奈米統計力學

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

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 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 . : a complex number and a function of time n : a set of quantum numbers : the probability associated with n N96770 微奈米統計力學

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. N96770 微奈米統計力學

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. total energy total number of particles np = 0, 1, 2, … for bosons np = 0, 1 for fermions level (energy eigenvalue) h : Planck’s constant N96770 微奈米統計力學

The levels p become continuous as the system volume V→∞. The spectrum can be divided into groups of levels containing g1, g2, g3, g4,… subcells. Each group is called a cell and has an average energy i. 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}. g4 g3 g2 g1 cell N96770 微奈米統計力學

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. N96770 微奈米統計力學

For Bosons Each of the gi 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

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

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

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

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

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

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

And equations of state for ideal Bose-Einstein gases become where 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 )