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
Top Related