2. 1 Chapter 9 Electron Spin and Pauli Principle §9.1 Electron Spin: Experimental evidences Double lines detected in experiments are in conflict with the theory of atomic spectroscopy. O.Stern and W. Gerlach, Z. Physik, 1922, 110, 9, 349. Existence of electron spin

3. 2 §9.2 Electron Spin: Theory 1928, Dirac, Relativistic Quantum Mechanics 1925 Uhlenbeck and Goudsmit Electron spin Electron spin Theorem Theorem 1: Electron has an intrinsic angular momentum, the relationship between the corresponding operators is similar with that of angular momentum operators.

4. 3 Theorem 2: For single electron, S z is connected with two eigenfunctions only, which correspond to eigenvalues of 1/2 ћ and -1/2 ћ, respectively. We denote the eigenfunction with eigenvalue of ½ ћ as , and the eigenfunction with eigenvalue of -1/2 ћ as . Obviously,  and  are not eigenfunctions of S x and S y

5. 4

6. 5 S S 图 10.1 电子自旋向量对 z 轴的两个可能取向

7. 6

8. 7 Theorem 3: Spinning electron can be taken as a tiny magnet, with magnetic momentum  Orbital motion Current I Theory of electric-magnetic field: A current around an area of A can be treated as a magnet with magnetic momentum IA/c Classic Quantum

9. 8 The Hamiltonian does not involve spin § 9.3 Spin and the Hydrogen Atom degeneracy For H atom

10. 9 § 9.4 The Pauli Princeple Identical particles: In classic mechanics: distinguishable In quantum chemistry: indistinguishable Permutation operator:

11. 10 Hamiltonian is symmetric with respect to the coordinates qs is also an eigenfunction of H with eigenvalue of E.

12. 11 symmetric antisymmetric

13. 12 Since the particles are indistinguishable, the eigenfunctions of P ij symmetric or antisymmetric Both wavefunctions correspond to the same state of the system.

14. 13 Pauli principle: The wave function of a system of electrons must be antisymmetric with respect to interchange of any two electrons. Half-integral spin: antisymmetric Fermions Integral spin: symmetric Bosons Pauli repulsion (not a real physical force)

15. 14 § 9.5 Ground State of the Helium Atom the ground state Ground state  (1)  (2)  (1)  (2)  (1)  (2)  (1)  (2)  (1)  (2)  (2)  (1) Sym None  (1)  (2)  (1)  (2)  (1)  (2)+  (2)  (1)  (1)  (2)-  (2)  (1) Sym A-Sym

16. 15 § 9.6 First excited state of the Helium Atom 1S(1)2S(2) and 2S(1)1S(2) 1/ 2 1/2 [1S(1)2S(2) + 2S(1)1S(2)] 1/ 2 1/2 [1S(1)2S(2) - 2S(1)1S(2)] Sym A-sym Triplet Singlet

17. 16 § 9.7 The Pauli Exclusion Principle Li

18. 17 sym antisym. impossible Li experimental:

19. 18 How to construct antisymmetric wavefunction with three functions? f, g, h: Orth-normalized functions f(1)g(2)h(3) P 12 P 13 P 23 f(2)g(1)h(3) f(3)g(2)h(1) f(1)g(3)h(2) P 12 f(3)g(1)h(2) P 12 f(2)g(3)h(1) Anti-symmetric wavefunction can be described as a linear combination of the functions above.

20. 19 Anti-symmetric requirement leads to: Normalization requirement leads to: Slater Determine

21. 20 How to construct antisymmetric wave function? Slater Det. Pauli exclusion priciple: Each spin-orbital can have only one electron. Spin-orbital

22. 21 求 的矩阵表示 § 9.8 Pauli Matrix

23. 22

24. 23 同理可求得其它表示矩阵

25. 24 Pauli 算符与 Pauli 矩阵