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 )

1. 參考資料 微奈米統計力學 Supplement on Quantum Statistical Mechanics 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) N96770 微奈米統計力學

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

3. General solution of Eq.(1) : Substitution of the boundary condition gives A = 0. Substitution of the boundary condition gives eigenvalue Ex : must also satisfy the normalization condition : * : conjugate ( Recall ) 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 : ……… (2) Periodic boundary conditions : 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 ……… (3) kB : Boltzmann constant where Z : partition function In the canonical ensemble, the partition function Zcan be written as ……… (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 ……… (6) Therefore, the density matrix can be expressed as N96770 微奈米統計力學

10. Once we have the partition function Z, the entropyS 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 : N96770 微奈米統計力學