Chapter 12

Quantum Mechanics

and Atomic Theory

Classical Physics

Classical mechanicsMaxwell’s theory of electricity,

magnetism and electro-magnetic radiation

ThermodynamicsKinetic 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，得出一黑體輻射公式，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

Tkc Bv 3

38

長波長 短波長

By combining the formulae of Wien and Rayleigh, Planck announced in October 1900 a formula now known as Planck's radiation formula.

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

1

2)(

3

3

kT

hv

e

v

c

hvR

h: Planck’s constantk: Boltzmann’s constantC: speed of lightv: frequency of light

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 large-scale 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

)2

( 2

hpx

The Atomic Spectrum of Hydrogen

The Bohr Model

Niel Bohr The Nobel Prize in Physics 1922

218

22.178 10 ( )

:

:

ZE J

nn 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

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ran operato:HamiltoniH

n)genfunctioorbital(eiction for ψ:wave fun

EψψH

equationeigenvalue

ˆ

ˆ

Particle in a box

m

kE

mEk

kxAmE

kxAkdx

kxAd

dx

xψd

kxAxψSuppose

hxψ

mE

dx

xψd

ψEψdx

d

mdx

d

mH

Vl energy)V(potentiaenergy)T(kinetic E

EψψH

2

2

)sin(2

)sin()sin()(

sin)(

)2

( )(2)(

)()(22

ˆ

)0(

ˆ

22

2

2

2

2

2

2

2

2

22

2

2

22

2

22

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.

1)(sin)(

1. bemust stategiven ain y probabilit

totalThe box. ain particle a finding ofy probabilit

relative themeansfunction wave theof square The

)sin()(

)sin()sin()(0)()0(

)sin()(

and constant thedefine

0

22

0

2

dxxL

nAdxxψ

xL

nAxψ

L

nknkL

nAkLALψLψψ

kxAxψ

Ak

LL

)sin(

2)(

)2

( 82

)(

2

1

2

1)

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4

1

2

1)(sin

2sin4

1

2

1sin

1

)(sin

2

2222

20

00

2

2

20

2

xL

n

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h

mL

hn

mL

n

E

LA

ALx

L

n

L

nxxdx

L

n

cxcxcxdx

Adxx

L

n

n

LLL

L

Three energy levels

Three Dimension Box

z)L

πn(y)

L

πn(x)

L

πn(

LLLz)yψ(x

) L

n

L

n

L

n (

m

hE

z

z

y

y

x

x

zyx

z

z

y

y

x

x

sinsinsin8

,,

8 2

2

2

2

2

22

Degenercy

　 　 　

221 212 122

　 　 　211 121 112

　 　 　　 111 　

E Lx=Ly=Lz

The spherical polar coordinate system

The Wave Function for the Hydrogen Atom

)(10178.2)8

(

)(ˆ

sin

1cot

112

2ˆ

2ˆˆˆ

ˆ

2

218

20

4

2

2

2

2

2222

2

22

22

22

2

n

ZJ

h

me

n

ZE

r

ZerVV

rrrrrrT

r

ZeVTH

EΨΨH

)R(r)Θ(r)Θ(Ψ

n

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 element2

1 1 1 12

2 2 2 2

[ ( , , )]

[ ( , , )]

r dv N

r dv N

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

The radial probability

distribution for the

hydrogen 1s orbital

in spherical shell

Bohr radius: 0.529Å

Denoted by a00.529Å

)4()

34

()( 22

3

22 rRdr

rdR

dr

dVR

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: the principal quantum number

l: 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 Letter Used

0 s

1 p

2 d

3 f

4 g

Quantum Numbers magnetic quantum number (ml)

Have integral values between l and –l, including zero.

Relates to the orientation in space of angular momentum associated with the orbital.

2Px

n value

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/molSecond 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 mecha

nical equations

2+ e-

e-

Zeff+

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

Penetration effect

Core electrons

Na:1s22s22p63s1

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 EnergyX(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)+e- I1=580 kJ/mol

Al+(g) →Al+2(g)+e- I2=1815 kJ/mol

Al+2(g) →Al+3 +e- I3=2740 kJ/mol

Al+3(g) →Al+4 +e- I4=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.