Chapter 35 Quantum Mechanics of Atoms

S-equation for H atom 2 Schrödinger equation for hydrogen atom: Separate variables:

Three quantum numbers (1) 3 1) Principal quantum number n : Energy is quantized, as same as Bohr theory 2) Orbital quantum number l : Solution is determined by 3 quantum numbers L is the magnitude of orbital angular momentum

Three quantum numbers (2) 4 L is also quantized, but in a different form! 3) Magnetic quantum number m l : Space quantization → L z < L ! Zeeman effect

The 4 th quantum number (1) 5 Stern-Gerlach experiment in 1921 : Ground state → l = 0 → magnetic moment μ = 0 G. E. Uhlenbeck and Goudsmit (1924): Except the orbital motion, the electron also has a spin and the spin angular momentum.

The 4 th quantum number (2) 6 Every elementary particle has a spin. Dirac: Spin is a relativistic effect Paul Dirac Nobel 1933 Spin quantum number can be: 1) Integers → boson, such as photon 2) Half-integers → fermion, such as electron

Possible states 7 Solution: Remember rules of quantum numbers Example1: How many different states are possible for an electron whose principal quantum number is n = 2 ? List all of them. nlmlml msms nlmlml msms 2001/2200-1/2 2111/2211-1/2 2101/2210-1/2 211/221-1/2

Energy and angular momentum 8 Solution: (a) n = 2, all states have same energy Example2: Determine (a) the energy and (b) the orbital angular momentum for each state in Ex1. (b) For l = 0: For l = 1: Macroscopic L → continuous

Wave function for H atom 9 The wave function for ground state: The probability density is: Radial probability distribution:

Electron cloud 10 There is no “orbit” for the electron in atom Probability distribution → “electron cloud”

Complex atoms 11 For complex atoms, atomic number Z > 1 Extra interaction → energy depend on n and l Two principles for the configuration of electrons 1) Lowest energy principle → ground state At the ground state of an atom, each electron tends to occupy the lowest energy level. Empirical formula of energy:

Pauli exclusion principle 12 Each electron occupies a state (n, l, m l, m s ) 2) Pauli exclusion principle: No two electrons in an atom can occupy the same quantum state. Wolfgang Pauli Nobel 1945 It is valid for all fermions How many electrons can be in state l = 0, 1, 2 ? How many electrons can be in state n = 1, 2, 3 ?

Shell structure of electrons 13 Electrons with same n → in the same shell l n spdfg s 2 22s 2 2p 6 33s 2 3p 6 3d 10 44s 2 4p 6 4d 10 4f 14 55s 2 5p 6 5d 10 5f 14 5g 18 with same n and l → same subshell

Periodic table of elements 14 From D. Mendeleev to quantum mechanics

Electron configurations 15 Solution: (a) Forbidden, only 2 allowed states in 2s Example3: Which of the following electron configurations are possible, and which forbidden? (a) 1s 2 2s 3 2p 3 ; (b) 1s 2 2s 2 2p 5 3s 2 ; (c) 1s 2 2s 2 2p 6 2d 2. (b) Allowed, but exited state. (c) Forbidden, no 2d subshell. Allowed configurations? (O) 1s 2 2s 2 2p 4 (Na) 1s 2 2s 2 2p 6 3s 1 (Mg) 1s 2 2s 2 2p 6 3s 2

*Lasers 16 “Light Amplification by Stimulated Emission of Radiation” → LASER Stimulated emission: Inverted population: Metastable state & optical pumping h f=E 2 -E 1 E2E2 E1E1

*Chapter 36 Molecules and Solids 17 This chapter should be studied by yourself Molecular spectra Bonding in solids Band theory of solids Semiconductors & diodes

The End Good luck!