Introduction to Quantum Mechanics (2017,1st semester)
Undergraduate course: # 400.307
Credit: 3
Class hour: Mon. / Wed. 15:30-16:45
Class room: 302-720
Lecturer: Prof. In Chung (Office Rm. 302-919, Tel: 880-7408, inching@snu.ac.kr )
Office Hour:
Teaching assistants: 이형석(Rm. 302-710, Tel: 880-1530, jabezlife@snu.ac.kr)
Textbooks: P.W. Atkins, Physical Chemistry, 10th ed. (7, 8, 9th ed. would be OK)
References: D.A. Mcquarrie, Physical Chemistry: A Molecular Approach
Evaluation
1) Two mid terms and Final (30% each)
2) Homework (5%), Attitude (5%)
4) Attendance: 3 points subtract from total points per absence.
3 absences = F. 2 late = 1 absence
* Cell phone use during class = immediate exit with 1 absence
Energy Materials Lab
Lecture schedule
Date
Lecture
Chapter
3/6, 3/8
Introduction to Quantum Theory
7
3/13, 3/15
Introduction to Quantum Theory
7
3/20, 3/22
Quantum Theory of Motion
8
3/27, 3/29
Quantum Theory of Motion
8
4/3, 4/5
No Class
-
4/8
Midterm #1 Exam (am 10:00)
-
4/10, 4/12
Atomic Structure and Spectra
9
4/17, 4/19
Atomic Structure and Spectra
9
4/24, 4/26
Atomic Structure and Spectra
9
5/1, 5/3
Molecular Structure
10
5/8, 5/10
Molecular Structure
10
5/13
Midterm #2 Exam (am 10:00)
-
5/15, 5/17
Molecular Structure
10
5/22, 5/24
Solids
18
5/29, 5/31
No Class
-
6/5, 6/7
Solids
18
6/10
Final Exam (am 10:00)
-
비고
No Class (5/3)
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Classical (Newton) Mechanics
1st law : Law of Inertia
2nd law : Equation of Motion
3rd law : Action-Reaction
𝐹Ԧ = 𝑚𝑎Ԧ = 𝑚𝑟Ԧሷ
𝐹Ԧ12 = −𝐹Ԧ21
Potential E vs. Force
𝑃1 (𝑟1 )
𝑑𝑤 = 𝐹Ԧ𝑒𝑥 ∙ 𝑑 𝑟Ԧ = −𝐹Ԧ ∙ 𝑑 𝑟Ԧ
𝑚
𝐹Ԧ : Force exerted to the particle by the system
𝐹Ԧ𝑒𝑥 : Force needed to move the particle in the direction of 1→2 at 𝑟Ԧ
𝐹Ԧ𝑒𝑥
Ԧ 𝑟)
Ԧ
If 𝐹Ԧ depends only on the position, i.e. 𝐹Ԧ = −𝐹(
𝐹Ԧ
2
𝑊12 = − න 𝐹Ԧ ∙ 𝑑 𝑟Ԧ = 𝑉2 − 𝑉1 = ∆𝑉
1
If 𝑟 = ∞
𝑃2 (𝑟2 )
2
∆𝑉 = − න 𝐹Ԧ ∙ 𝑑 𝑟Ԧ = 𝑉2 − 𝑉∞
Take 𝑉∞ = 0 as reference
ර 𝐹Ԧ ∙ 𝑑 𝑟Ԧ = 0
∞
Path independent
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Conservative Field
𝑚
𝑉2
Energy conservation
𝑉2 − 𝑉1 = 𝐸𝑘 1 − 𝐸𝑘 2
𝑉1 + 𝐸𝑘 1 = 𝑉2 + 𝐸𝑘 2
𝐸 𝑡𝑜𝑡𝑎𝑙 = 𝐸𝑘 + 𝑉
ℎ2
𝑉1
ℎ1
Kinetic Energy
Potential Energy
Nonconservative Field
Damping (frictional) force
Falling body in the air
𝐹𝑓𝑟𝑖𝑐 ≈ 𝛾𝑣
𝐹𝑔𝑟𝑎𝑣 = 𝑚𝑔
Ԧ 𝑟,
𝐹Ԧ = 𝐹(
Ԧ 𝑣)
Ԧ
∆V = ∆𝐸𝑘 + ∆𝐸 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑙𝑜𝑠𝑠
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Trajectory : in terms of force
𝐹Ԧ = 𝑚𝑎Ԧ = 𝑚𝑟Ԧሷ
𝐹𝑥 𝑥, 𝑦, 𝑧 = 𝑚𝑥ሷ
𝐹𝑦 𝑥, 𝑦, 𝑧 = 𝑚𝑦ሷ
𝐹𝑧 𝑥, 𝑦, 𝑧 = 𝑚𝑧ሷ
Solution : 𝑟Ԧ = 𝑟(𝑡;
Ԧ
𝑟𝑜 , 𝑟0ሶ )
𝑟0ሶ (𝑡 = 0)
𝑟𝑜 (𝑡 = 0)
Coupled
2nd order
D.E.
Trajectory
𝑚
𝑟Ԧ (𝑡 > 0)
𝑟Ԧ (𝑡 < 0)
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Trajectory : in terms of energy
𝐸 = 𝐸𝑘 + 𝑉 𝑥, 𝑦, 𝑧 =
1
𝑚𝑣 2 + 𝑉
2
For simplicity, take 1D case as an example
𝑝2
𝐸 = 𝐸𝑘 + 𝑉 𝑥 =
+ 𝑉(𝑥)
2𝑚
𝑑𝑥
2{𝐸 − 𝑉(𝑥)}2
=𝑉=
𝑑𝑡
𝑚
(1)
Solution : 𝑥 = 𝑥(𝑡; 𝑥𝑜 )
𝑝 = 𝑝(𝑡) can be calculated from eq. (1)
also called trajectory
∴ Two description of trajectory are equivalent.
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Vibrational Motion
Harmonic oscillator
Hook’s law (within elastic limit)
𝐹 = −𝑘𝑥
𝑥
1 2
𝑉 𝑥 = න −𝐹𝑑𝑥 = 𝑘𝑥
2
0
Where 𝑉 0 = 0 as reference
1
1 2
2
𝐸 = 𝑚𝑣 + 𝑘𝑥
2
2
Equation of Motion
𝐹 = −𝑘𝑥 = 𝑚𝑥ሷ
𝑘
𝑥ሷ + 𝑥 = 0
𝑚
Initial condition
𝑥ሷ +
𝜔2 𝑥
𝑘
= 0 (𝜔 = )
𝑚
2
𝑥 0 = 𝐴 (𝐴 : Amplitude)
𝑥ሶ 0 = 0
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Solution
𝑥 = 𝐴 sin 𝜔𝑡 𝑥ሶ = 𝐴𝜔 cos 𝜔𝑡
1
1 2 2
1 2
2
2
𝐸𝑘 = 𝑚𝑣 = 𝜔 𝐴 cos 𝜔𝑡 = 𝑘𝐴 cos 2𝜔𝑡
2
2
2
1 2 1 2
𝑉 = 𝑘𝑥 = 𝑘𝐴 sin 2𝜔𝑡
2
2
1 2
𝐸 = 𝐸𝑘 + 𝑉 = 𝑘𝐴
2
𝐸(𝑡𝑜𝑡𝑎𝑙) ∝ 𝐴2
E can take any (+) value
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Allowed Energy Values in Classic Mechanics
Ex. A constant force applied to a body initially at rest in free space
𝑉 = 0 ( everywhere )
𝑑𝑝
=𝐹 0≤𝑡≤𝜏
𝑑𝑡
𝑡>𝜏
=0
𝑝 = 0 : initial condition
𝑡
𝐹(𝑡)
𝐹
τ
𝑡
𝑝 𝑡 = න 𝐹𝑑𝑡 = 𝑝 0 + 𝐹𝑡 (0 < 𝑡 < 𝜏)
0
𝑝2
𝐹2𝜏2
𝐸=
=
2𝑚
2𝑚
Translational energy can take any non-negative real value.
Continuous energy spectrum
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Rotational Motion
Translation
Rotation
𝐹 = 𝑚𝑎
2
1
𝑝
𝐸𝑘 = 𝑚𝑣 2 =
2
2𝑚
𝑇 = 𝐼𝛼
1 2 𝐽2
𝐸𝑘 = 𝐼𝜔 =
2
2𝐼
𝑇 : Torque
𝐼 : Moment of inertia
𝛼 : Angular acceleration
𝜔 : Angular velocity
𝐽 : Angular momentum
𝑇
𝐼
𝛼
𝜔
𝐽
𝐹
𝑚
𝑎
𝑣
𝑝
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Pure Rotation
𝜔
O
𝐽Ԧ
𝑣Ԧ
𝑟Ԧ
𝜃
𝜔 = 𝜃ሶ 𝜔
ෝ
𝑣Ԧ = 𝑟Ԧ × 𝜔
𝐽Ԧ = 𝑟Ԧ × 𝑝Ԧ
𝑑 𝐽Ԧ 𝑑
𝑇=
= (𝑟Ԧ × 𝑝)
Ԧ
𝑑𝑡 𝑑𝑡
𝑑
=𝑚
𝑟Ԧ × 𝑣Ԧ = 𝑚 𝑣Ԧ × 𝑣Ԧ + 𝑚𝑟Ԧ × 𝑣Ԧሶ
𝑑𝑡
= 𝑟Ԧ × 𝑝Ԧሶ = 𝑟Ԧ × 𝐹Ԧ
1
1
1
1 2
2
2
2
𝐸 = 𝑚𝑣 = 𝑚 𝑟Ԧ × 𝜔 ∙ 𝑟Ԧ × 𝜔 = 𝑚𝑟 𝜔 = 𝐼𝜔
2
2
2
2
𝑝2
𝑟 2 𝑝2
(𝑟Ԧ × 𝑝)
Ԧ ∙ (𝑟Ԧ × 𝑝)
Ԧ
𝐽2
𝐸=
=
=
=
2
2𝑚 2𝑚𝑟
2𝐼
2𝐼
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Pure Rotation
Constant torque applied to a body initially at rest for 0 ≤ 𝑡 ≤ 𝜏
𝜔
O
𝐽Ԧ
𝑣Ԧ
𝑟Ԧ
𝜃
𝐽 𝑡 = 𝑇𝜏 + 𝐽(0)
(0 ≤ 𝑡 ≤ 𝜏)
(𝑡 > 𝜏)
𝐽2 𝑇 2 𝜏 2
𝐸=
=
2𝐼
2𝐼
𝐸 can take any 𝑡 real value.
𝐽 𝑡 = 𝑇𝜏
Continuous rotational energy spectrum
V 𝑥
Characteristics of Classic Mechanics
𝐸
1. Deterministic,
Trajectory 𝑟Ԧ = 𝑟(𝑡;
Ԧ 𝑟𝑜 , 𝑟0ሶ )
Past
Present
Future
2. Continuous Energy Spectrum
𝐾𝐸
𝑃𝐸
−𝐴
𝐴
𝑥
3. Dynamic variables (𝑟,
Ԧ 𝑝,
Ԧ 𝐸 , etc.) takes a specific value.
No limitation in measurement accuracy.
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Equipartition Theorem
 Useful conclusion from Classical Mechanics
 Energy distribution among various modes of molecular motion
Molecular motion
Translation
(x, y, z direction)
Rotation
(around 2 axis)
Vibration
How much energy does a molecule take in various forms of
motion in thermal equilibrium at Temp. T?
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Classical Mechanics Conclusion:
1
Each quadratic term in the energy expression takes 2 𝑘𝑇 on the average
3
1
2
2
2
 Translation : 𝐸 =
𝑘𝑇
(𝑝 + 𝑝𝑦 + 𝑝𝑧 )
2
2𝑚 𝑥
1
𝐽2
 Rotation : Monoatomic molecule 𝐸 =
𝑘𝑇
2
2𝐼
1
Diatomic molecule
𝐸 = (𝐽𝑥 2 + 𝐽𝑦 2 + 𝐽𝑧 2 )
2𝐼
2
𝑝
1 2
 Vibration : 𝐸 =
+ 𝑘𝑥
𝑘𝑇
2𝑚 2
𝑘𝑇
Ex. 𝐶𝑣 of monoatomic ideal gas
3
𝐸 = 𝑘𝑇
2
3
𝑑𝐸
3
𝐶
=
𝑅
𝐶𝑣 =
= 𝑘 𝑣
2
𝑑𝑇
2
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Boltzmann Distribution Law
𝑝(𝐸) ∝ 𝑒 −𝐸/𝑘𝑇
Ex. 1-D Translation
𝑝 𝐸 =
∞
Thermal Equilibrium
Fundamental law
1
𝐸 = 𝑚𝑣 2
2 1 2
𝑝(𝐸) ∝ 𝑒 −2𝑚𝑣 /𝑘𝑇
𝑒
1
−2𝑚𝑣 2 /𝑘𝑇
∞ −1𝑚𝑣 2 /𝑘𝑇
‫׬‬−∞ 𝑒 2
∞
=
1
𝑚/2𝜋𝑘𝑇
1
− 𝑚𝑣 2 /𝑘𝑇
𝑒 2
1
2𝑘𝑇
1
1
−2𝑚𝑣 2 /𝑘𝑇
2
𝐸 = න 𝐸 ∙ 𝑝 𝐸 𝑑𝐸 =
න
𝑚𝑣 𝑒
𝑑𝑣 = 𝑘𝑇
𝑚 −∞ 2
2
0
∞
∞
𝜋
𝜋
2
2
−𝑥
, න 𝑥 2 𝑒 −𝑥 𝑑𝑥 =
)
(Cf. න 𝑒
𝑑𝑥 =
2
4
−∞
−∞
You will see later that equipartition theorem does not hold true in
Quantum Mechanics. However, it is still true in a certain limit.
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
Failure of Classical Mechanics
 At the end of 19C
 Results of fundamental experiments
 Microscopic system related phenomena
I.
II.
III.
IV.
V.
VI.
Could not be interpreted by which was
based on Classical Mechanics
Black body Radiation
Heat Capacity of Solid
Photoelectric Effect
Spectrum of H atom
Compton Scattering
Electron Diffraction from Solid Surface
Great challenge to Classical Mechanics
Bold Assumption needed to for breakthrough
Interpretation of each experimental result contributed to the birth of Quantum Mechanics
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 hot object emit electromagnetic radiation
e.g. iron bar : As temperature is increased, red  yellow  blue  …
( T wavelength , frequency  )
The energy distribution in a black-body cavity at several temperatures. Note how the
energy density increases in the visible region as the temperature is raised, and how the
peak shifts to shorter wavelengths. The total energy density (the area under the curve)
increases as the temperature is increased (as T4).
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 Black-body : ideal emitter, perfect absorber & perfect emitter
 Many times absorption & emission
Thermal equilibrium at temperature T
Leaking out through pinhole
 T  color shifts toward the blue
An experimental representation of a black
body is a pinhole in an otherwise closed
container. The radiation is reflected many
times within the container and comes to
thermal equilibrium with the walls at a
temperature T. Radiation leaking out
through the pinhole is characteristic of
the radiation within the container.
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation

Wien’s law (Wien’s displacement law, 1893)
Tmax = const. = 1/5c2, c2 = 1.44 cmK
(max : maximum distribution wavelength at T)
e.g. At 1000 K  max ~ 2900 nm
Sunlight peak at ~500 nm → T = 5800 K
 Stefan-Boltzmann law (1879)
 Expression 1) : Total energy density , ( = E/V, radiation energy per unit volume)
𝜀 = 𝑎𝑇 4
 Expression 2) Excitance M, (radiation power per unit surface, the brightness of the
emission)
𝑀 = 𝑇4
(𝜎 : Stefan-Boltzmann constant, 𝜎 = 5.67 x 10-8 W·m-2K-4 )
 e.g. At 1000K, 1 cm2 surface radiate about 6 W (cf. W = J/s)
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 Rayleigh-Jeans law
 On the 19th Century, classical approach by Rayleigh
 Electromagnetic field as a collection of a oscillators of all possible frequencies
Energy distribution calculation from mean energy 𝐸
= 𝑘𝑇 for each oscillator
The electromagnetic vacuum can be
regarded as able to support oscillations of the
electromagnetic field. When a high-frequency
short-wavelength oscillator (a) is excited, that
frequency of radiation is present. The
presence of low-frequency long-wavelength
radiation (b) signifies that an oscillator of the
corresponding frequency has been excited.
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
1
2
 At temperature T, Average energy = 𝑘𝑇
1) Translation : 𝐸 =
2) Rotation :
3) Vibration :
1
2𝑚
3
𝑝𝑥 2 + 𝑝𝑦 2 + 𝑝𝑧 2 = 2 𝑘𝑇
Linear
𝐽2
1
𝐸 = = 𝑘𝑇
2𝐼
2
2
𝑝
1
𝐸 = 2𝑚 + 2 𝑘𝑥 2 = 𝑘𝑇
momentum
 Population on thermal equilibrium at temperature T
𝑝 𝐸 = 𝐴𝑒 −𝐸/𝑘𝑇
“Boltzmann distribution law”
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 Rayleigh-Jeans law
𝑑 = 𝑑,  =
8 𝐸
4
𝑑 =
8𝑘𝑇
4
𝑑
 : energy density,  : proportionality constant,
k : Boltzmann constant (1.381 x 10-23 J·K-1)
 = 𝑐/ ,  = 𝑐 
𝑑 = −𝑐(𝑑/2)  𝑑 = −2𝑑/𝑐
𝑑 = (8 2𝑘𝑇/𝑐3)𝑑
 Quite successful at long 
 but it fails at lower  (UV, X-rays…) : The Rayleigh-Jeans law predicts an infinite
energy density at short wavelengths.
“Ultraviolet catastrophe”(자외선 파탄)
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 Why  = (8<E>/4)d ?
𝐿
λ
𝐿=𝑛
, 𝑛 = 0, 1, 2
2
2𝐿
2𝐿
λ=
, 𝑛=
𝑛
λ
𝑖𝑓) 𝐿 ≫ λ, 𝑛 can be expressed by the function of λ ( 𝑛 λ )
Number of oscillators from λ to λ + 𝑑λ :
𝑑𝑛 = −
Expansion from
1-D to 3-D
2𝐿
𝑑λ
2
λ
Number of oscillators in a unit volume from λ to λ + 𝑑λ :
Let) 𝐸 : Average energy in each oscillator
𝑑𝑛 =
8π
𝑑λ
4
λ
Then,
𝟖𝝅
𝑑𝜀 = 𝐸 𝑑𝑛 = 𝟒 𝑬 𝒅𝝀 ≡ 𝝆
𝝀
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 By classical mechanics, even cool objects should radiate in the visible and UV regions
no darkness even at low T (?)
 The Planck distribution
 In 1900, Max Planck proposed each oscillator is not continuous  “Energy quantization”
 proposing that the energy of each oscillator is limited to discrete values and cannot be
varied arbitrarily
cf. classical mechanics: all possible energies are allowed
𝐸 = ℎν, 2ℎν, 3ℎν,……(integer multiples of h)
𝐸 = 𝑛ℎν, 𝑛 = 0, 1, 2, 3, ….
ℎ : Planck constant, 6.626 x 10-34 Js
λν = 𝑐 , : wavelength, : frequency
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 Classical mechanics
Average energy 𝐸
∞
𝐸 =
Put
‫׬‬0 𝐸 · 𝑝 𝐸 𝑑𝐸
∞
‫׬‬0 𝑝
𝐸 𝑑𝐸
1
−
= 𝑎,
𝑘𝑇
−𝐸/𝑘𝑇
, 𝑝 𝐸 = 𝐴𝑒
𝑎𝑥
𝑎𝑥 ∞
𝑒
𝑒
use න 𝑒 𝑎𝑥 𝑑𝑥 =
න
, 𝑥𝑒 𝑎𝑥 𝑑𝑥 = 2 (𝑎𝑥 − 1)
𝑎
𝑎 0
0
∞
∞
∞
𝐸
𝐴 𝑎𝐸
−𝑘𝑇 ∞
 Denominator : න 𝑝 𝐸 𝑑𝐸 = 𝐴 න
= 𝑒 = −𝑘𝑇𝐴 · 𝑒
ฬ = 𝑘𝑇 · 𝐴
𝑎
0
0
0
∞
∞
𝐸
𝐸
∞
−
𝑎𝐸
2
𝑘𝑇
න
𝐸
·
𝑝
𝐸
𝑑𝐸
=
𝐴
න
𝐸
·
𝑒
𝑑𝐸
=
𝐴
·
(𝑘𝑇)
·
𝑒
(−
−
1)
ฬ
 Numerator :
𝑘𝑇
0
0
0
𝑒 𝑎𝐸 𝑑𝐸
= (𝑘𝑇)2 · 𝐴
∴
(𝑘𝑇)2 · 𝐴
𝐸 =
= 𝑘𝑇
𝑘𝑇 · 𝐴
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
 Planck distribution
Average energy 𝐸
∞
𝐸 =
𝐸 =
‫׬‬0 𝐸 · 𝑝 𝐸 𝑑𝐸
∞
‫׬‬0 𝑝 𝐸 𝑑𝐸
𝐴(0 + ℎν ·
−𝐸/𝑘𝑇 𝐸 = 𝑛ℎν (𝑛 = 0, 1, 2, 3, … )
,
, 𝑝 𝐸 = 𝐴𝑒
ℎ𝜈
−𝑘𝑇
𝑒
𝐴(1 +
ℎ𝜈
−𝑘𝑇
𝑒
+ 2ℎν ·
+
2ℎ𝜈
− 𝑘𝑇
𝑒
2ℎ𝜈
− 𝑘𝑇
𝑒
+ ⋯)
,
+ ⋯)
(if) 𝑒 −ℎν/𝑘𝑇 = α < 1, 1 + α + α2 + ⋯ =
1
1−α
 Numerator : ℎν α + 2α2 + 3α3 + ⋯ = ℎν · α 1 + 2α + 3α2 + ⋯
𝑑
𝑑
α
α + α2 + α3 + ⋯ = ℎν · α ·
(
)
𝑑α
𝑑α 1 − α
1· 1−α +α
αℎν
= ℎν · α ·
=
(1 − α)2
(1 − α)2
1
 Denominator :
1−α
= ℎν · α ·
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
ℎνα/(1 − α)2
α
𝑒 −ℎν/𝑘𝑇
1
ℎ𝑐
1
𝐸 =
= ℎν
= ℎν
=
ℎν
=
(
)
1/(1 − α)
(1 − α)
1 − 𝑒 −ℎν/𝑘𝑇
λ 𝑒 ℎ𝑐/λ𝑘𝑇 − 1
𝑒 ℎν/𝑘𝑇 − 1
𝑐
(∵ ν = )
λ
8𝜋 𝐸
𝑑ε = ρ𝑑λ =
λ4
∴ ρ=
dλ
8𝜋ℎ𝑐
1
λ5 𝑒 ℎ𝑐/λ𝑘𝑇 − 1
“The Planck distribution accounts very well for the experimentally determined
distribution of radiation. Planck's quantization hypothesis essentially quenches the
contributions of high-frequency, short-wavelength oscillators. The distribution
coincides with the Rayleigh-Jeans distribution at long wavelengths.”
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(1) Black body radiation
ℎ𝑐
→∞
λ𝑘𝑇
ℎ𝑐
1
𝑒 λ𝑘𝑇 ≫ 5
λ
 Short λ (λ → 0)
∴ 𝜌=0
(Difference from Rayleigh-Jeans distribution)
ℎ𝑐
≪1
λ𝑘𝑇
 Long λ
∴ 𝜌=

𝑑𝜌
𝑑λ
ℎ𝑐
𝑒 λ𝑘𝑇
8𝜋ℎ𝑐 λ𝑘𝑇 8𝜋
·
= 4 𝑘𝑇
λ5
ℎ𝑐
λ
=0
∞
Find out𝜆𝑚𝑎𝑥
∞
 𝜀 = ‫׬‬0 𝑑ε = ‫׬‬0 𝜌 𝑑𝜆
−1= 1+
ℎ𝑐
ℎ𝑐
+⋯ −1≈
λ𝑘𝑇
λ𝑘𝑇
“Rayleigh-Jeans Law”
(Coincidence with classical mechanics)
“Wien’s Law”
“Stefan-Boltzmann Law”
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(2) Heat capacity
 Dulong & Petit’s law (19 C)
 Monatomic solid
: 𝐸 = 𝑘𝑇 for each direction  3𝑘𝑇 for 3-D
 N atoms
: molar internal energy 𝑈𝑚 = 3𝑁𝐴𝑘𝑇 = 3𝑅𝑇
 Constant volume heat capacity
: 𝐶𝑉
1)
𝐶𝑉,𝑚
= (𝑈𝑚/𝑇)𝑉 = 3𝑅 (= 24.9 JK-1mol-
 Deviation at low temperature
3𝑅
𝑂
,𝑚
: 𝑇  0  𝐶𝑉
𝑇 (𝐾)
,𝑚
0
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(2) Heat capacity
 Einstein formula (1905)
 all the atoms oscillate with the same frequency  at low T, few oscillators
possess energy to oscillate; T , enough energy for all the oscillators
 Using Planck’s hypothesis (𝐸 = 𝑛ℎ)
: all 3𝑁 atomic oscillators  vibrational energy of crystal: 3𝑁 𝐸
1
Energy levels of the harmonic oscillators 𝜀𝑛 = ℎν𝐸 (𝑛 + 2)
(𝑛 = 0, 1, 2, … )
∞
෍ 𝑥 −𝑛 =
Assuming the oscillators are in thermal equilibrium at temp. T,
𝑛=0
1
1−𝑥
Partition function for a single oscillator
∞
∞
∞
1
𝑒 −𝛽ℎν𝐸 /2
−𝛽ℎν𝐸 /2
−𝑛𝛽ℎν𝐸
=
෍𝑒
 𝑞 = ෍ exp[−𝛽 𝜀𝑛 ] = ෍ exp[−𝛽 ℎν𝐸 (𝑛 + 2) 𝑛 ] = 𝑒
1 − 𝑒 −𝛽ℎν𝐸
𝑛=0
𝑛=0
𝑛=0
𝑑𝑙𝑛𝑞
𝑑 𝛽ℎν𝐸
ℎν𝐸
ℎν𝐸
𝑢=
=
+ ln(1 − 𝑒 −𝛽ℎν𝐸 ) =
+ 𝛽ℎν
Mean energy per oscillator
𝑑𝛽
𝑑𝛽
2
2
𝑒 𝐸−1
Zero-point energy
Energy of 3N oscillators in the N-atom solid
𝑈 = 3𝑁𝑢 = 3𝑁
ℎν𝐸
ℎν𝐸
+ 𝛽ℎν
2
𝑒 𝐸 −1
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(2) Heat capacity
put 𝜃𝐸 =
𝜕𝑈
𝐶𝑉 =
𝜕𝑇
ℎν
𝑘
𝑉
𝜃𝐸 𝑒 𝜃𝐸 /2𝑇
𝑓=
𝑇 𝑒 𝜃𝐸 /𝑇 − 1
 “Einstein temperature”
𝜕𝑈
= 3𝑁
𝜕𝛽
𝜕𝛽
(𝜃𝐸 /𝑇)2 𝑒 𝜃𝐸 /𝑇 ,
= 3𝑁𝑘 𝜃 /𝑇
𝜕𝑇
(𝑒 𝐸 − 1)2
𝑉
𝐶𝑉,𝑚
𝜃𝐸
= 3𝑅
𝑇
2
𝑒 𝜃𝐸 /𝑇
= 3𝑅𝑓 2
𝜃
/𝑇
2
𝐸
(1 − 𝑒
)
(1) At high T (𝑇 ≫ 𝜃𝐸 )
𝜃𝐸
𝑓=
𝑇
1+
𝜃𝐸
+⋯
2𝑇
𝜃
1+ 𝐸 +⋯ −1
𝑇
∴ 𝑪𝑽,𝒎 = 𝟑𝑹
≅1
(Same result in Classical Mechanics
Dulong & Petit’s Law)
(2) At low T (𝑇 ≪ 𝜃𝐸 )
𝜃𝐸 𝑒 𝜃𝐸 /2𝑇
𝜃𝐸 −𝜃 /2𝑇
𝑓≅
=
·𝑒 𝐸
𝑇 𝑒 𝜃𝐸 /𝑇
𝑇
1
→∞
𝑒 −𝜃𝐸 /2𝑇 → 0 More rapidly than
𝑇
∴ 𝑻 → 𝟎, 𝒇 → 𝟎  𝑪𝑽,𝒎 → 𝟎
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(2) Heat capacity
Experimental low-temperature molar
heat capacities and the temperature
dependence predicted on the basis of
Einstein's
theory.
His
equation
accounts for the dependence fairly
well, but is everywhere too low.
- still poor in experimental data since Einstein assumed all the atoms oscillate with
the same frequency
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(2) Heat capacity
 Debye formula
 Consider to oscillate   0 to D
(in real crystal atoms are coupled by the interatomic forces and do not oscillate independently)
Debye's modification of Einstein's
calculation gives very good
agreement with experiment. For
copper, D = 2 corresponds to
about 170 K, so the detection of
deviations from Dulong and Petit's
law had to await advances in lowtemperature physics.
 Quantization must be introduced in order to explain thermal properties of solids
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(3) Photoelectric Effect
1902 Lenard: the electron energy were entirely independent of the light intensity.
Further, there was a certain threshold frequency below which no photoelectron were
ejected, no matter how bright the light beam.
Albert Einstein (1905) showed that
the puzzle of photoelectric effect are
easily explained once the illuminating
radiation is a collection of particles
(photons)
𝒒𝑽
= 𝒉
–
𝑷
(kinetic energy (energy of the (work to get out
of electron) incoming photon) of the metal)
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(3) Photoelectric Effect
In the photoelectric effect, it is
found that no electrons are
ejected when the incident
radiation has a frequency
below a value characteristic of
the metal and, above that
value, the kinetic energy of the
photoelectrons varies linearly
with the frequency of the
incident radiation.
 Einstein (1905-6)
 < 0 : (threshold ): no emission, even at strong radiation intensity
 > 0 : electron emission even at very low intensity
 Kinetic energy of ejected electron  , independent of radiation intensity
if h >  (work function): electron emission,
1
𝑚𝑒𝑣2 = ℎ − 
2
(: the energy required to remove an electron from the metal to infinity)
 Energy depended on the frequency of the incident light  nh
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(3) Photoelectric Effect
Kinetic energy of
ejected electron
Energy needed to
remove electron
from metal
Energy supplied
by photon
The photoelectric effect can be explained if it is supposed that the incident radiation is
composed of photons that have energy proportional to the frequency of the radiation. (a) The
energy of the photon is insufficient to drive an electron out of the metal. (b) The energy of the
photon is more than enough to eject an electron, and the excess energy is carried away as
the kinetic energy of the photoelectron (the ejected electron).
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(4) Atomic & Molecular spectra
1862 A. J. Angstrom (1814-74) : hydrogen
What do these lines mean? All this was very puzzling!
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(4) Atomic & Molecular spectra
 spectrum: radiation absorbed or emitted by atoms & molecules
A region of the spectrum of
radiation emitted by excited
iron atoms consists of
radiation at a series of discrete
wavelengths (or frequencies).
 radiation is emitted or absorbed at a series of discrete frequencies
 energy of atoms/molecules is confined to discrete values
 Energy is quantized
 Only explain hydrogen (one-electron) spectra
1) fail to explain the spectra of atoms more than one electron
2) incorrect to regard the electrons in atoms as discrete particles with precise positions and velocities
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(4) Atomic & Molecular spectra
Spectral lines can be accounted for if we
assume that a molecule emits a photon as
it changes between discrete energy levels.
Note that high-frequency radiation is
emitted when the energy change is large.
 Balmer (1885): visible spectrum of atomic hydrogen
1/ = 𝑅𝐻(1/22 – 1/𝑛2), n = 3, 4,…;
(Empirical expression)
𝑅𝐻 : Rydberg const. (= 1.09678 x 105 cm-1)
 Lyman: UV series, 1/ = 𝑅𝐻(1/12 – 1/𝑛2)
 Paschen: IR, 1/ = 𝑅𝐻(1/32 – 1/𝑛2)
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(4) Atomic & Molecular spectra
 Why lines?  Bohr (1913) : Planck quantum hypothesis + classical mechanics
 Bohr’s hypothesis
1) Electron exists in a discrete set of stable, stationary orbits in the atom
cf) perfect orbit : different from that in quantum mechanics
(𝑳 = 𝒓 𝒙 𝒑)
2) Angular momentum of orbital : quantized
𝐿 = 𝑟 𝑝 = 𝑚𝑒 𝑣𝑒 𝑟
𝑚𝑒 𝑣𝑒 𝑟 =
3) Transition between orbits :
𝑛ℎ
2𝜋
∆𝐸 = ℎν

Frequency ν = ∆𝐸/ℎ
4) Dynamical equilibrium between proton and electron electrostatic attraction force
 Centripetal force
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(4) Atomic & Molecular spectra
From (Coulombic Force) = (Centripetal Force)
𝑚𝑒 𝑣 2
1 𝑒2
=
𝑟𝑒
4𝜋𝜀𝑜 𝑟𝑒 2
1
𝑒2
 𝑟𝑒 = 4𝜋𝜀 ∙ 𝑚 𝑣 2
𝑜
𝑒
𝑛ℎ
𝐿 = 𝑟Ԧ × 𝑝Ԧ = 𝑟𝑒 𝑚𝑒 𝑣 =
𝑛 = 1,2, …
2𝜋
𝑛ℎ
Substitute 𝑣 with
2𝜋𝑟𝑒 𝑚𝑒
1
𝑒2
1
𝑒 2 4𝜋 2 𝑟𝑒 2 𝑚𝑒 2
𝑟𝑒 =
∙
=
∙
∙
4𝜋𝜀𝑜 𝑚𝑒 𝑣 2 4𝜋𝜀𝑜 𝑚𝑒
𝑛2 ℎ2
1
1
1
𝑚𝑣 2 = 𝑚𝑟 2 𝜔2 = 𝐼𝜔2
2
2
2
𝑣 = 𝑟𝜔
𝑣
𝐿 = 𝐼𝜔 = 𝑚𝑟 2 ∙ = 𝑚𝑣𝑟
𝑟

𝑛2 ℎ2 𝜀𝑜
𝑟𝑒 = 2
𝑒 𝑚𝑒 𝜋

𝑟𝑒 = 𝑎𝑜 ≅ 0.53 Å Radius of atom = “Bohr radius”
1
1 𝑒2
2
𝐸 = 𝐸𝐾 + 𝑉 = 𝑚𝑒 𝑣 −
2
4𝜋𝜀𝑜 𝑟𝑒
2
1 𝑒
1 𝑒2
1 𝑒2
=
∙ −
∙
=−
∙
8𝜋𝜀𝑜 𝑟𝑒 4𝜋𝜀𝑜 𝑟𝑒
8𝜋𝜀𝑜 𝑟𝑒
1
𝑒 2 𝑚𝑒 𝜋
𝑚𝑒 𝑒 4 1
2
=−
∙𝑒 ∙ 2 2 =−
∙
8𝜋𝜀𝑜
𝑛 ℎ 𝜀𝑜
8𝜀𝑜 2 ℎ2 𝑛2
𝑛=1
𝑚𝑒 → 𝜇
(Reduced mass =
∆𝐸
μ𝑒 4
ν=
=−
ℎ
8𝜀𝑜 2 ℎ2
𝑚𝑒 𝑚𝑝
𝑚𝑒 +𝑚𝑝
)
1
1
1
1
−
=
−𝑅
−
𝐻
𝑛2 𝑛′2
𝑛2 𝑛′2
μ𝑒 4
𝑅𝐻 =
8𝜀𝑜 2 ℎ2
Energy Materials Lab
Chapter. 7. Introduction to Quantum Theory
(4) Atomic & Molecular spectra
 Bohr’s postulates
𝑛ℎ
2𝜋
 Radiation is only emitted when an atom makes translations between stationary states
 Quantized angular momentum : 𝐿 = 𝑟𝑝 = 𝑟𝑒 𝑚𝑒 𝑣 =
𝐸𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑝ℎ𝑜𝑡𝑜𝑛 = 𝐸𝑚 − 𝐸𝑛
 From Bohr’s postulates
ℎ2 𝜀𝑜
ℎ2 𝜀𝑜 2
2
( 𝑎𝑜 = Bohr radius; 𝑎𝑜 = 2
)
𝑟𝑛 = 2
𝑛 = 𝑎𝑜 𝑛
𝑒 𝑚𝑒 𝜋
𝑒 𝑚𝑒 𝜋
ℎ 1
ℎ
1
and 𝑝𝑛 =
𝑣𝑛 =
2𝑎𝑜 𝜋 𝑛
2𝑎𝑜 𝑚𝑒 𝜋 𝑛
ℎ
1
 We can rewrite the above as :
= 2𝜋𝑎𝑜 𝑛 = 2𝜋𝑟𝑛
𝑝𝑛
𝑛
 Then quantization of momentum implies that the circumference of the allowed
states is and integer multiple of the de Broglie wavelength λ𝑑𝐵 = ℎ/𝑝
𝑛λ𝑑𝐵 = 2𝜋𝑟𝑛
Energy Materials Lab