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Chapter 35
Quantum Mechanics of
Atoms
S-equation for H atom
Schrödinger equation for hydrogen atom:

2
2m
 2 
Separate variables:
e2
4 0 r
  E
 (r, ,  )  R(r )( )( )
1 d 2 dR
2m
e2
l (l  1)
(r
)  [ 2 (E 
)
]R  0
2
2
dr
4 o r
r dr
r
ml 2
1 d
dΘ
(sin
)  [l (l  1)  2 ]Θ  0
sin  d

sin 
d 2Φ
2

m
l Φ 0
2
d
2
Three quantum numbers (1)
Solution is determined by 3 quantum numbers
1) Principal quantum number n :
13.6eV
En 
, n  1, 2, ...
2
n
Energy is quantized, as same as Bohr theory
2) Orbital quantum number l :
L  l (l  1)  , l  0, 1, ..., n  1
L is the magnitude of orbital angular momentum
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Three quantum numbers (2)
L  l (l  1)  , l  0, 1, ..., n  1
L is also quantized, but in a different form!
3) Magnetic quantum number ml :
Lz  ml , ml  l , ..., l
Space quantization → Lz < L !
Zeeman effect
n2
l 1
ml  1
0
1
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The 4th quantum number (1)
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.
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The 4th quantum number (2)
Every elementary particle has a spin.
Dirac: Spin is a relativistic effect
Spin quantum number can be:
Paul Dirac
1) Integers → boson, such as photon
Nobel 1933
2) Half-integers → fermion, such as electron
1
s ,
2
3
S  s(s  1)  
,
2
1
ms  
2
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Possible states
Example1: How many different states are
possible for an electron whose principal quantum
number is n = 2 ? List all of them.
Solution: Remember rules of quantum numbers
n
2
2
2
l
0
1
1
ml
0
1
0
ms
1/2
1/2
1/2
n
2
2
2
l
0
1
1
ml
0
1
0
ms
-1/2
-1/2
-1/2
2
1
-1
1/2
2
1
-1
-1/2
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Energy and angular momentum
Example2: Determine (a) the energy and (b) the
orbital angular momentum for each state in Ex1.
Solution: (a) n = 2, all states have same energy
13.6eV
E2 
 3.4eV
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(b) For l = 0:
L  l (l  1)   0
For l = 1: L  l (l  1)   2  1.5 1034 J  s
Macroscopic L → continuous
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Wave function for H atom
The wave function for ground state:
 100 

1
 r03
e
r
r0
h 2 0
Bohr radius r0 
 me2
 100
The probability density is:
2
Radial probability distribution:
 dV    4 r 2 dr  Pr dr
2
2
 Pr  4 r 
2
2
2
4r
 3 e
r0

2r
r0
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Electron cloud
There is no “orbit” for the electron in atom
Probability distribution
→ “electron cloud”
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Complex atoms
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:
E
n  0.7l
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Pauli exclusion principle
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.
It is valid for all fermions
Wolfgang Pauli
Nobel 1945
How many electrons can be in state l = 0, 1, 2 ?
How many electrons can be in state n = 1, 2, 3 ?
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Shell structure of electrons
Electrons with same n → in the same shell
with same n and l → same subshell
s
0
p
1
d
2
1
2
3
4
1s2
2s2
3s2
4s2
2p6
3p6
4p6
3d10
4d10
5
5s2
5p6
5d10
l
n
f
3
g
4
4f14
5f14
5g18
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Periodic table of elements
From D. Mendeleev to quantum mechanics
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Electron configurations
Example3: Which of the following electron
configurations are possible, and which forbidden?
(a) 1s22s32p3 ; (b) 1s22s22p53s2; (c) 1s22s22p62d2.
Solution: (a) Forbidden, only 2 allowed states in 2s
Allowed configurations?
(O) 1s22s22p4
(b) Allowed, but exited state.
(c) Forbidden, no 2d subshell.
(Na) 1s22s22p63s1
(Mg) 1s22s22p63s2
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*Lasers
“Light Amplification by Stimulated Emission of
Radiation” → LASER
Stimulated emission:
Inverted population:
N high  N low
E2
hf=E2 -E1
E1
Metastable state & optical pumping
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*Chapter 36 Molecules and Solids
This chapter should be studied by yourself
Molecular spectra
Bonding in solids
Band theory of solids
Semiconductors & diodes
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