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

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.

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

4

5 S S 图 10.1 电子自旋向量对 z 轴的两个可能取向

6

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

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

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

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

11 symmetric antisymmetric

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

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)

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

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

16 § 9.7 The Pauli Exclusion Principle Li

17 sym antisym. impossible Li experimental:

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.

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

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

21 求 的矩阵表示 § 9.8 Pauli Matrix

22

23 同理可求得其它表示矩阵

24 Pauli 算符与 Pauli 矩阵