N96770 微奈米統計力學 1 上課時間 ： 上課地點 : 國立成功大學工程科學系越生講堂 (41X01 教室 ) 參考資料 微奈米統計力學 Supplement on Quantum Statistical Mechanics

N96770 微奈米統計力學 2 Free Particle in a Box First consider a particle m that translates freely between two walls of distance L : Schrödinger equation of the system :  : wave function of the particle E x : energy of the system Ĥ : Hamiltonian operator h : Planck’s constant Periodic boundary conditions : L x m ……… (1)

N96770 微奈米統計力學 3 Substitution of the boundary condition gives A = 0. General solution of Eq.(1) : Substitution of the boundary condition gives eigenvalue E x :  must also satisfy the normalization condition : ( Recall ) * : conjugate

N96770 微奈米統計力學 4 Now consider the particle m that translates freely inside a cubic box with length L : Schrödinger equation of the system : Periodic boundary conditions : ……… (2)

N96770 微奈米統計力學 5 Similarly the eigenfunction (wave function for this case) of Eq.(2) is And the eigenvalue E is

N96770 微奈米統計力學 6 Without loss of generality, the wave function can be written as r : position vector k : wave vector where The eigenvalue E can be also written as

N96770 微奈米統計力學 7 Being a canonical ensemble, the density matrix  can be written as k B : Boltzmann constant where Z : partition function In the canonical ensemble, the partition function Z can be written as ……… (3) ……… (4)

N96770 微奈米統計力學 8 Substituting  E and E into Eq.(3), we have The energy levels become continuous as the system volume V→∞, so the summation can be approximated by the integral : ……… (5)

N96770 微奈米統計力學 9 The following integral formula is used to obtain the result of Eq.(5) : Similarly, Eq.(4) can be obtained as Therefore, the density matrix can be expressed as ……… (6)

N96770 微奈米統計力學 10 Once we have the partition function Z, the entropy S can be obtained by the following procedures : 1. First recall the Helmhotz free energy A : Substitute Eq.(6) into A and use T in terms of  : 2. Calculate entropy from the thermodynamics relationship between A and S :