Chapter 35

Quantum Mechanics of

Atoms

S-equation for H atom

2

Schrödinger equation for hydrogen atom:2 2

2

02 4

eE

m r

Separate variables: ( , , ) ( ) ( ) ( ) r R r

22

2 2 2

1 2 ( 1)( ) [ ( ) ] 0

4

o

d dR m e l lr E R

dr dr rr r2

2

1(sin ) [ ( 1) ] 0

sin sin

lmd dΘ

l l Θd

22

20

l

d Φm Φ

d

Three quantum numbers (1)

3

1) Principal quantum number n :

2

13.6, 1, 2, ...

n

eVE n

n

Energy is quantized, as same as Bohr theory

2) Orbital quantum number l :

Solution is determined by 3 quantum numbers

( 1) , 0, 1, ..., 1 L l l l n

L is the magnitude of orbital angular momentum

Three quantum numbers (2)

4

( 1) , 0, 1, ..., 1 L l l l n

L is also quantized, but in a different form!

3) Magnetic quantum number ml :

, , ..., z l lL m m l l

Space quantization → Lz < L !

Zeeman effect2n

1l

1lm0

1

The 4th 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 4th 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

1,

2s

3( 1) ,

2 S s s

1

2sm

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.

n l ml ms n l ml ms

2 0 0 1/2 2 0 0 -1/2

2 1 1 1/2 2 1 1 -1/2

2 1 0 1/2 2 1 0 -1/2

2 1 -1 1/2 2 1 -1 -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.

2

13.63.4

4

eVE eV

(b) For l = 0: ( 1) 0 L l l

For l = 1: ( 1) 2 L l l 341.5 10 J s

Macroscopic L → continuous

Wave function for H atom

9

The wave function for ground state:

0100 3

0

1

r

rer

20

0 2Bohr radius

h

rme

The probability density is:2

100

Radial probability distribution:

2 2 24 rdV r dr P dr

0

2222

30

4 4

r

rr

rP r e

r

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: 0.7E n l

Pauli exclusion principle

12

Each electron occupies a state (n, l, ml , ms)

2) Pauli exclusion principle:

No two electrons in an atom can

occupy the same quantum state.

Wolfgang PauliNobel 1945It 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

s p d f g

0 1 2 3 4

1 1s2

2 2s2 2p6

3 3s2 3p6 3d10

4 4s2 4p6 4d10 4f14

5 5s2 5p6 5d10 5f14 5g18

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) 1s22s32p3 ; (b) 1s22s22p53s2; (c) 1s22s22p62d2.

(b) Allowed, but exited state.

(c) Forbidden, no 2d subshell.

Allowed configurations? (O) 1s22s22p4

(Na) 1s22s22p63s1

(Mg) 1s22s22p63s2

*Lasers

16

“Light Amplification by Stimulated Emission of Radiation” → LASER

Stimulated emission:

Inverted population:

Metastable state & optical pumping

h f=E2 -E1

E2

E1high lowN N

*Chapter 36 Molecules and Solids

17

This chapter should be studied by yourself

Molecular spectra

Bonding in solids

Band theory of solids

Semiconductors & diodes