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Chapter 12
Quantum Mechanics
and Atomic Theory
Classical Physics
 Classical
mechanics
 Maxwell’s theory of electricity,
magnetism and electro-magnetic
radiation
 Thermodynamics
 Kinetic theory
Eletromagnetic Radiation
Class radiation
Blackbody Radiation
Ultraviolet Catastrophe(紫外線崩潰)
The classical theory of matter, which
assumes that matter can absorb or emit
any quantity of energy, predicts a
radiation profile that has no maximum
and goes to infinite intensity at very
short wavelengths.
古典理論解釋失敗
1900年，J.W.Rayleigh，J.H.Jeans根據
古典電動力學和統計物理理論，得出
8 3
 v  3 k BT
一黑體輻射公式，
c
即Rayleigh-Jeans law。此公式只在低
頻部分與實驗曲線比較符合，高頻部
分是發散的，與實驗明顯不符，即
ultraviolet catastrophe
By combining the formulae of Wien and Rayleigh,
Planck announced in October 1900 a formula now
known as Planck's radiation formula.
長波長
短波長
Max Planck


1858~1947
Planck initiated the
study of quantum
mechanics when he
announced in 1900 his
theoretical research
into the radiation and
absorption of
heat/light by a black
body.
Max Planck’s Theory
The Nobel Prize in Physics
1918
3
2h
R (v )  3 
c
v
e
hv
kT
h: Planck’s constant
k: Boltzmann’s constant
C: speed of light
v: frequency of light
1
Planck’s Impacts
In classical physics, energy is a
continuous variable.
 Planck defined the amount of energy, a
quantum of energy, ∆E=nh
 In quantum physics, the energy of a
system is quantized.

Albert Einstein


1879~1955
Einstein contributed more than
any other scientist to the
modern vision of physical
reality. His special and general
theories of relativity are still
regarded as the most
satisfactory model of the largescale universe that we have.
Photoelectric Effect
Albert Einstein,
The Nobel Prize in Physics 1921
KEelectron=1/2mv2=hv-hv0
hv: energy of incident photon
hv0: energy required to remove electron
from metal’s surface
The wave nature of electrons
Louis de Broglie,
The Nobel Prize in
Physics 1929
光具有物質與波雙重性質
λ=h/mv
Light waves
Werner Heisenberg
1901~1976
 Werner
Heisenberg did
important work in
Quantum
Mechanics as well
as nuclear physics.

Uncertainty Principle
Werner Karl Heisenberg
The Nobel Prize in Physics 1932
任何一個粒子無法將位置與動量同時
很精確的量測出來

h
xp 
( 
)
2
2
The Atomic Spectrum
of Hydrogen
The Bohr Model
Niel Bohr
The Nobel Prize in Physics 1922
E  2.178 10
18
2
Z
J( 2 )
n
n : integer
Z : atomic number
 The
electron in a
hydrogen atom moves
around the nucleus only
in certain allowed
circular orbits.
量子發展如同瞎子摸象
 Max
Planck－能量不連續
 Albert Einstein－能階概念
 Louis de Broglie－粒子波動
 Werner Heisenberg－粒子運動測不
準
 Niel Bohr－光波不連續
Quantum Mechanics
 By
the mid-1920s, Werner Karl
Heisenberg, Louis de Broglie and
Erwin SchrÖdinger developed the
wave mechanics or more commonly,
quantum mechanics.
Tunnel Effect
Quantum effect in biological systems
 Rudolph A.
Marcus was awarded the
1992 Nobel Prize in Chemistry.
 Marcus theory
 Electron transfer reactions
SchrÖdinger Equation
eigenvalue equation
Hˆ ψ  Eψ
ψ:wave function for orbital(eigenfunction)
Hˆ :Hamiltonian operator
E: total energy(eigenvalue)
Particle in a box
Hˆ ψ  Eψ
E  T(kinetic energy)  V(potential energy) (V  0)
2
2
2
2

d

d
Hˆ  

(ψ )  E (ψ )
2
2
2m dx
2m dx
d 2ψ ( x)
2mE
h

  2 ψ ( x ) ( 
)
2
dx

2
Suppose ψ ( x)  A sin kx
d 2ψ ( x) d 2 ( A sin kx )
2mE
2

  k ( A sin kx )   2 ( A sin kx )
2
2
dx
dx

2 2
2
mE

k
2
 k   2  E 

2m
Boundary Conditions
 The
particle cannot be outside the
box-it is bound inside the box.
 In a given state the total probability
of finding the particle in the box
must be 1.
 The wave function must be
continuous.
define the constant k and A
ψ ( x)  A sin( kx )
ψ (0)  ψ ( L)  0  ψ ( L)  A sin( kL )  A sin( n )
n
n
kL  n  k 
 ψ ( x)  A sin(
x)
L
L
The square of the wave function means the relative
probabilit y of finding a particle in a box. The total
probabilit y in a given state must be 1.
L
L
2
2
2 n
0 ψ ( x)dx  0 A sin ( L x)dx  1
n
1
0 sin ( L x)dx  A2
1
1
2
  sin cxdx  x  c sin 2cx
2
4
L
2

L
0
n
1 L 1 n
2 n
1
1
sin ( )xdx  x 0 
sin(
x)  L  2
L
2
4 L
L
2
A
0
L
2
2
 A
L
2 n 2
 ( )
2 2
n
h
h
L
En 

( 
)
2
2m
8mL
2
2
n
ψ ( x) 
sin(
x)
L
L
Three energy levels
Three Dimension Box
2
2
x
2
x
2
y
2
y
n
2
z
2
z
n
h
n
E
(


)
8m L L
L
ψ(x, y, z) 
nyπ
nx π
8
nz π
sin (
x)sin (
y) sin (
z)
Lx Ly Lz
Lx
Ly
Lz
Degenercy
E
221
212
122
211
121
112
111
Lx=Ly=Lz
The spherical polar coordinate system
The Wave Function for the Hydrogen
Atom
Ψ  R(r)Θ(r)Θ( )
Hˆ Ψ  EΨ
2
2

Ze
Hˆ  Tˆ  Vˆ  
2  
2
r
 2 2  1 2
1

1
2 
 2
 2 cot 
 2 2
 2

2

r
r

r
r


r


r
sin





2
Ze
Vˆ  V (r )  
r
2
Z 2 me4
Z
18
E n  2 (
)


2
.
178

10
J( 2 )
2
n 8 0 h
n
2

Tˆ  
2
The Physical Meaning of a Wave
Function
The square of the function evaluated at a
particular point in space indicates the
probability of finding an electron near that
point.
Ψ2:probability distribution
dv: small volume element

[(r1 , 1 , 1 )] dv N1

2
[(r2 ,  2 , 2 )] dv N 2
2
The probability
distribution for the
hydrogen 1s orbital
Calculate the probability
at points along a line
drawn outward in any
direction from nucleus.
number of nodes=n-1
nodes
0.529Å
The radial probability
distribution for the
hydrogen 1s orbital
in spherical shell
Bohr radius: 0.529Å
Denoted by a0
4 3
d ( r )
dV
R2 ( )  R2 3
 R 2 (4r 2 )
dr
dr
Relative Orbital Size
 The
normally accepted arbitrary
definition of the size of the hydrogen
1s orbital is the radius of the sphere
that encloses 90% of the total electron
probability.
 For H(1s), r(1s)=2.6a0=1.4Å
Quantum Numbers
 n:
 l:
the principal quantum number
the angular momentum quantum
number
 Ml:
the magnetic quantum number
Quantum Numbers
principal quantum number (n)




Have integral values (1,2,3…)
It is related to the size and energy of orbital.
As n increases, the orbital becomes larger and
the electron spends more time farther from the
nucleus
An increase in n also means higher energy
Quantum Numbers
angular momentum quantum number (l)


Have integral values from 0 to n-1.
Determines the shapes of the atomic orbital.
Value
0
1
2
3
4
Letter Used
s
p
d
f
g
Quantum Numbers
magnetic quantum number (ml)
integral values between l and –l,
including zero.
 Relates to the orientation in space of
angular momentum associated with
the orbital.
 Have
n value
2Px
l Value
Orientation in space
Electron Spin and Pauli Principle
 electron
spin quantum number (ms):
+1/2 and -1/2
 Pauli Principle: In a given atom, no
two electrons can have the same set of
four quantum numbers (n, l, ml, ms)
Polyelectronic Atoms
the kinetic energy of the electrons as they
move around the nucleus
 the potential energy of attraction between
the nucleus and the electrons
 the potential energy of repulsion between
the two electrons
 First ionization energy: 2372 kJ/mol
 Second ionization energy: 5248 kJ/mol

Hydrogen Like Approximation



Zeff=Zactual-(effect of electron repulsions)
1<Zeff<2
Substituting Zeff in place of Z=1 in the hydrogen
wave mechanical equations
2+
e-
Zeff+
e-
e-
Actual He atom
Hypothetical He atom
Valence electrons and Core
electrons

Valence electrons: the electrons in the outermost
principal quantum level of an atom.

Core electrons: inner electrons
Na:1s22s22p63s1
Core electrons
Penetration effect
K: 1s22s22p63s23p64s1
Aufbau principle




The physical and chemical properties of elements
is determined by the atomic structure.
The atomic structure is, in turn, determined by the
electrons and which shells, subshells and orbitals
they reside in.
The rules of placing electrons within shells is
known as the Aufbau principle.
As protons are added one by one to the nucleus to
build up the elements, electrons are similarly
added to these atomic orbitals.
Ionization Energy
X(g) →X+(g)+e


The ionization energy for a particular electron
in an atom is a source of information about the
energy of the orbital it occupies in the atom.
Resulting ion will not reorganize in response to
the removal of an electron.
The ionization energies do provide information
that is quite useful in testing the orbital model
of the atom.
Ionization of Al
Al(g) →Al+(g)+eI1=580 kJ/mol
Al+(g) →Al+2(g)+e- I2=1815 kJ/mol
Al+2(g) →Al+3 +eI3=2740 kJ/mol
Al+3(g) →Al+4 +eI4=11600 kJ/mol
 (1) 1s22s22p63s23p1 →1s22s22p63s2
 (2) 1s22s22p63s2 → 1s22s22p63s1
 (3) 1s22s22p63s1→1s22s22p6
 (4) 1s22s22p6→ 1s22s22p5
The values of first ionization energy for
the elements in the first five periods.
○
○

As n increases, the size
of the orbital increases,
and the electron is easier
to remove.
The electron affinity for atoms among the first
20 elements that form stable, isolated X- ions.