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Raffi Hovasapian

Raffi Hovasapian

The Particle in a Box Part I

Slide Duration:

Table of Contents

I. Classical Thermodynamics Preliminaries
The Ideal Gas Law

46m 5s

Intro
0:00
Course Overview
0:16
Thermodynamics & Classical Thermodynamics
0:17
Structure of the Course
1:30
The Ideal Gas Law
3:06
Ideal Gas Law: PV=nRT
3:07
Units of Pressure
4:51
Manipulating Units
5:52
Atmosphere : atm
8:15
Millimeter of Mercury: mm Hg
8:48
SI Unit of Volume
9:32
SI Unit of Temperature
10:32
Value of R (Gas Constant): Pv = nRT
10:51
Extensive and Intensive Variables (Properties)
15:23
Intensive Property
15:52
Extensive Property
16:30
Example: Extensive and Intensive Variables
18:20
Ideal Gas Law
19:24
Ideal Gas Law with Intensive Variables
19:25
Graphing Equations
23:51
Hold T Constant & Graph P vs. V
23:52
Hold P Constant & Graph V vs. T
31:08
Hold V Constant & Graph P vs. T
34:38
Isochores or Isometrics
37:08
More on the V vs. T Graph
39:46
More on the P vs. V Graph
42:06
Ideal Gas Law at Low Pressure & High Temperature
44:26
Ideal Gas Law at High Pressure & Low Temperature
45:16
Math Lesson 1: Partial Differentiation

46m 2s

Intro
0:00
Math Lesson 1: Partial Differentiation
0:38
Overview
0:39
Example I
3:00
Example II
6:33
Example III
9:52
Example IV
17:26
Differential & Derivative
21:44
What Does It Mean?
21:45
Total Differential (or Total Derivative)
30:16
Net Change in Pressure (P)
33:58
General Equation for Total Differential
38:12
Example 5: Total Differential
39:28
II. Energy
Energy & the First Law I

1h 6m 45s

Intro
0:00
Properties of Thermodynamic State
1:38
Big Picture: 3 Properties of Thermodynamic State
1:39
Enthalpy & Free Energy
3:30
Associated Law
4:40
Energy & the First Law of Thermodynamics
7:13
System & Its Surrounding Separated by a Boundary
7:14
In Other Cases the Boundary is Less Clear
10:47
State of a System
12:37
State of a System
12:38
Change in State
14:00
Path for a Change in State
14:57
Example: State of a System
15:46
Open, Close, and Isolated System
18:26
Open System
18:27
Closed System
19:02
Isolated System
19:22
Important Questions
20:38
Important Questions
20:39
Work & Heat
22:50
Definition of Work
23:33
Properties of Work
25:34
Definition of Heat
32:16
Properties of Heat
34:49
Experiment #1
42:23
Experiment #2
47:00
More on Work & Heat
54:50
More on Work & Heat
54:51
Conventions for Heat & Work
1:00:50
Convention for Heat
1:02:40
Convention for Work
1:04:24
Schematic Representation
1:05:00
Energy & the First Law II

1h 6m 33s

Intro
0:00
The First Law of Thermodynamics
0:53
The First Law of Thermodynamics
0:54
Example 1: What is the Change in Energy of the System & Surroundings?
8:53
Energy and The First Law II, cont.
11:55
The Energy of a System Changes in Two Ways
11:56
Systems Possess Energy, Not Heat or Work
12:45
Scenario 1
16:00
Scenario 2
16:46
State Property, Path Properties, and Path Functions
18:10
Pressure-Volume Work
22:36
When a System Changes
22:37
Gas Expands
24:06
Gas is Compressed
25:13
Pressure Volume Diagram: Analyzing Expansion
27:17
What if We do the Same Expansion in Two Stages?
35:22
Multistage Expansion
43:58
General Expression for the Pressure-Volume Work
46:59
Upper Limit of Isothermal Expansion
50:00
Expression for the Work Done in an Isothermal Expansion
52:45
Example 2: Find an Expression for the Maximum Work Done by an Ideal Gas upon Isothermal Expansion
56:18
Example 3: Calculate the External Pressure and Work Done
58:50
Energy & the First Law III

1h 2m 17s

Intro
0:00
Compression
0:20
Compression Overview
0:34
Single-stage compression vs. 2-stage Compression
2:16
Multi-stage Compression
8:40
Example I: Compression
14:47
Example 1: Single-stage Compression
14:47
Example 1: 2-stage Compression
20:07
Example 1: Absolute Minimum
26:37
More on Compression
32:55
Isothermal Expansion & Compression
32:56
External & Internal Pressure of the System
35:18
Reversible & Irreversible Processes
37:32
Process 1: Overview
38:57
Process 2: Overview
39:36
Process 1: Analysis
40:42
Process 2: Analysis
45:29
Reversible Process
50:03
Isothermal Expansion and Compression
54:31
Example II: Reversible Isothermal Compression of a Van der Waals Gas
58:10
Example 2: Reversible Isothermal Compression of a Van der Waals Gas
58:11
Changes in Energy & State: Constant Volume

1h 4m 39s

Intro
0:00
Recall
0:37
State Function & Path Function
0:38
First Law
2:11
Exact & Inexact Differential
2:12
Where Does (∆U = Q - W) or dU = dQ - dU Come from?
8:54
Cyclic Integrals of Path and State Functions
8:55
Our Empirical Experience of the First Law
12:31
∆U = Q - W
18:42
Relations between Changes in Properties and Energy
22:24
Relations between Changes in Properties and Energy
22:25
Rate of Change of Energy per Unit Change in Temperature
29:54
Rate of Change of Energy per Unit Change in Volume at Constant Temperature
32:39
Total Differential Equation
34:38
Constant Volume
41:08
If Volume Remains Constant, then dV = 0
41:09
Constant Volume Heat Capacity
45:22
Constant Volume Integrated
48:14
Increase & Decrease in Energy of the System
54:19
Example 1: ∆U and Qv
57:43
Important Equations
1:02:06
Joule's Experiment

16m 50s

Intro
0:00
Joule's Experiment
0:09
Joule's Experiment
1:20
Interpretation of the Result
4:42
The Gas Expands Against No External Pressure
4:43
Temperature of the Surrounding Does Not Change
6:20
System & Surrounding
7:04
Joule's Law
10:44
More on Joule's Experiment
11:08
Later Experiment
12:38
Dealing with the 2nd Law & Its Mathematical Consequences
13:52
Changes in Energy & State: Constant Pressure

43m 40s

Intro
0:00
Changes in Energy & State: Constant Pressure
0:20
Integrating with Constant Pressure
0:35
Defining the New State Function
6:24
Heat & Enthalpy of the System at Constant Pressure
8:54
Finding ∆U
12:10
dH
15:28
Constant Pressure Heat Capacity
18:08
Important Equations
25:44
Important Equations
25:45
Important Equations at Constant Pressure
27:32
Example I: Change in Enthalpy (∆H)
28:53
Example II: Change in Internal Energy (∆U)
34:19
The Relationship Between Cp & Cv

32m 23s

Intro
0:00
The Relationship Between Cp & Cv
0:21
For a Constant Volume Process No Work is Done
0:22
For a Constant Pressure Process ∆V ≠ 0, so Work is Done
1:16
The Relationship Between Cp & Cv: For an Ideal Gas
3:26
The Relationship Between Cp & Cv: In Terms of Molar heat Capacities
5:44
Heat Capacity Can Have an Infinite # of Values
7:14
The Relationship Between Cp & Cv
11:20
When Cp is Greater than Cv
17:13
2nd Term
18:10
1st Term
19:20
Constant P Process: 3 Parts
22:36
Part 1
23:45
Part 2
24:10
Part 3
24:46
Define : γ = (Cp/Cv)
28:06
For Gases
28:36
For Liquids
29:04
For an Ideal Gas
30:46
The Joule Thompson Experiment

39m 15s

Intro
0:00
General Equations
0:13
Recall
0:14
How Does Enthalpy of a System Change Upon a Unit Change in Pressure?
2:58
For Liquids & Solids
12:11
For Ideal Gases
14:08
For Real Gases
16:58
The Joule Thompson Experiment
18:37
The Joule Thompson Experiment Setup
18:38
The Flow in 2 Stages
22:54
Work Equation for the Joule Thompson Experiment
24:14
Insulated Pipe
26:33
Joule-Thompson Coefficient
29:50
Changing Temperature & Pressure in Such a Way that Enthalpy Remains Constant
31:44
Joule Thompson Inversion Temperature
36:26
Positive & Negative Joule-Thompson Coefficient
36:27
Joule Thompson Inversion Temperature
37:22
Inversion Temperature of Hydrogen Gas
37:59
Adiabatic Changes of State

35m 52s

Intro
0:00
Adiabatic Changes of State
0:10
Adiabatic Changes of State
0:18
Work & Energy in an Adiabatic Process
3:44
Pressure-Volume Work
7:43
Adiabatic Changes for an Ideal Gas
9:23
Adiabatic Changes for an Ideal Gas
9:24
Equation for a Fixed Change in Volume
11:20
Maximum & Minimum Values of Temperature
14:20
Adiabatic Path
18:08
Adiabatic Path Diagram
18:09
Reversible Adiabatic Expansion
21:54
Reversible Adiabatic Compression
22:34
Fundamental Relationship Equation for an Ideal Gas Under Adiabatic Expansion
25:00
More on the Equation
28:20
Important Equations
32:16
Important Adiabatic Equation
32:17
Reversible Adiabatic Change of State Equation
33:02
III. Energy Example Problems
1st Law Example Problems I

42m 40s

Intro
0:00
Fundamental Equations
0:56
Work
2:40
Energy (1st Law)
3:10
Definition of Enthalpy
3:44
Heat capacity Definitions
4:06
The Mathematics
6:35
Fundamental Concepts
8:13
Isothermal
8:20
Adiabatic
8:54
Isobaric
9:25
Isometric
9:48
Ideal Gases
10:14
Example I
12:08
Example I: Conventions
12:44
Example I: Part A
15:30
Example I: Part B
18:24
Example I: Part C
19:53
Example II: What is the Heat Capacity of the System?
21:49
Example III: Find Q, W, ∆U & ∆H for this Change of State
24:15
Example IV: Find Q, W, ∆U & ∆H
31:37
Example V: Find Q, W, ∆U & ∆H
38:20
1st Law Example Problems II

1h 23s

Intro
0:00
Example I
0:11
Example I: Finding ∆U
1:49
Example I: Finding W
6:22
Example I: Finding Q
11:23
Example I: Finding ∆H
16:09
Example I: Summary
17:07
Example II
21:16
Example II: Finding W
22:42
Example II: Finding ∆H
27:48
Example II: Finding Q
30:58
Example II: Finding ∆U
31:30
Example III
33:33
Example III: Finding ∆U, Q & W
33:34
Example III: Finding ∆H
38:07
Example IV
41:50
Example IV: Finding ∆U
41:51
Example IV: Finding ∆H
45:42
Example V
49:31
Example V: Finding W
49:32
Example V: Finding ∆U
55:26
Example V: Finding Q
56:26
Example V: Finding ∆H
56:55
1st Law Example Problems III

44m 34s

Intro
0:00
Example I
0:15
Example I: Finding the Final Temperature
3:40
Example I: Finding Q
8:04
Example I: Finding ∆U
8:25
Example I: Finding W
9:08
Example I: Finding ∆H
9:51
Example II
11:27
Example II: Finding the Final Temperature
11:28
Example II: Finding ∆U
21:25
Example II: Finding W & Q
22:14
Example II: Finding ∆H
23:03
Example III
24:38
Example III: Finding the Final Temperature
24:39
Example III: Finding W, ∆U, and Q
27:43
Example III: Finding ∆H
28:04
Example IV
29:23
Example IV: Finding ∆U, W, and Q
25:36
Example IV: Finding ∆H
31:33
Example V
32:24
Example V: Finding the Final Temperature
33:32
Example V: Finding ∆U
39:31
Example V: Finding W
40:17
Example V: First Way of Finding ∆H
41:10
Example V: Second Way of Finding ∆H
42:10
Thermochemistry Example Problems

59m 7s

Intro
0:00
Example I: Find ∆H° for the Following Reaction
0:42
Example II: Calculate the ∆U° for the Reaction in Example I
5:33
Example III: Calculate the Heat of Formation of NH₃ at 298 K
14:23
Example IV
32:15
Part A: Calculate the Heat of Vaporization of Water at 25°C
33:49
Part B: Calculate the Work Done in Vaporizing 2 Mols of Water at 25°C Under a Constant Pressure of 1 atm
35:26
Part C: Find ∆U for the Vaporization of Water at 25°C
41:00
Part D: Find the Enthalpy of Vaporization of Water at 100°C
43:12
Example V
49:24
Part A: Constant Temperature & Increasing Pressure
50:25
Part B: Increasing temperature & Constant Pressure
56:20
IV. Entropy
Entropy

49m 16s

Intro
0:00
Entropy, Part 1
0:16
Coefficient of Thermal Expansion (Isobaric)
0:38
Coefficient of Compressibility (Isothermal)
1:25
Relative Increase & Relative Decrease
2:16
More on α
4:40
More on κ
8:38
Entropy, Part 2
11:04
Definition of Entropy
12:54
Differential Change in Entropy & the Reversible Path
20:08
State Property of the System
28:26
Entropy Changes Under Isothermal Conditions
35:00
Recall: Heating Curve
41:05
Some Phase Changes Take Place Under Constant Pressure
44:07
Example I: Finding ∆S for a Phase Change
46:05
Math Lesson II

33m 59s

Intro
0:00
Math Lesson II
0:46
Let F(x,y) = x²y³
0:47
Total Differential
3:34
Total Differential Expression
6:06
Example 1
9:24
More on Math Expression
13:26
Exact Total Differential Expression
13:27
Exact Differentials
19:50
Inexact Differentials
20:20
The Cyclic Rule
21:06
The Cyclic Rule
21:07
Example 2
27:58
Entropy As a Function of Temperature & Volume

54m 37s

Intro
0:00
Entropy As a Function of Temperature & Volume
0:14
Fundamental Equation of Thermodynamics
1:16
Things to Notice
9:10
Entropy As a Function of Temperature & Volume
14:47
Temperature-dependence of Entropy
24:00
Example I
26:19
Entropy As a Function of Temperature & Volume, Cont.
31:55
Volume-dependence of Entropy at Constant Temperature
31:56
Differentiate with Respect to Temperature, Holding Volume Constant
36:16
Recall the Cyclic Rule
45:15
Summary & Recap
46:47
Fundamental Equation of Thermodynamics
46:48
For Entropy as a Function of Temperature & Volume
47:18
The Volume-dependence of Entropy for Liquids & Solids
52:52
Entropy as a Function of Temperature & Pressure

31m 18s

Intro
0:00
Entropy as a Function of Temperature & Pressure
0:17
Entropy as a Function of Temperature & Pressure
0:18
Rewrite the Total Differential
5:54
Temperature-dependence
7:08
Pressure-dependence
9:04
Differentiate with Respect to Pressure & Holding Temperature Constant
9:54
Differentiate with Respect to Temperature & Holding Pressure Constant
11:28
Pressure-Dependence of Entropy for Liquids & Solids
18:45
Pressure-Dependence of Entropy for Liquids & Solids
18:46
Example I: ∆S of Transformation
26:20
Summary of Entropy So Far

23m 6s

Intro
0:00
Summary of Entropy So Far
0:43
Defining dS
1:04
Fundamental Equation of Thermodynamics
3:51
Temperature & Volume
6:04
Temperature & Pressure
9:10
Two Important Equations for How Entropy Behaves
13:38
State of a System & Heat Capacity
15:34
Temperature-dependence of Entropy
19:49
Entropy Changes for an Ideal Gas

25m 42s

Intro
0:00
Entropy Changes for an Ideal Gas
1:10
General Equation
1:22
The Fundamental Theorem of Thermodynamics
2:37
Recall the Basic Total Differential Expression for S = S (T,V)
5:36
For a Finite Change in State
7:58
If Cv is Constant Over the Particular Temperature Range
9:05
Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure
11:35
Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure
11:36
Recall the Basic Total Differential expression for S = S (T, P)
15:13
For a Finite Change
18:06
Example 1: Calculate the ∆S of Transformation
22:02
V. Entropy Example Problems
Entropy Example Problems I

43m 39s

Intro
0:00
Entropy Example Problems I
0:24
Fundamental Equation of Thermodynamics
1:10
Entropy as a Function of Temperature & Volume
2:04
Entropy as a Function of Temperature & Pressure
2:59
Entropy For Phase Changes
4:47
Entropy For an Ideal Gas
6:14
Third Law Entropies
8:25
Statement of the Third Law
9:17
Entropy of the Liquid State of a Substance Above Its Melting Point
10:23
Entropy For the Gas Above Its Boiling Temperature
13:02
Entropy Changes in Chemical Reactions
15:26
Entropy Change at a Temperature Other than 25°C
16:32
Example I
19:31
Part A: Calculate ∆S for the Transformation Under Constant Volume
20:34
Part B: Calculate ∆S for the Transformation Under Constant Pressure
25:04
Example II: Calculate ∆S fir the Transformation Under Isobaric Conditions
27:53
Example III
30:14
Part A: Calculate ∆S if 1 Mol of Aluminum is taken from 25°C to 255°C
31:14
Part B: If S°₂₉₈ = 28.4 J/mol-K, Calculate S° for Aluminum at 498 K
33:23
Example IV: Calculate Entropy Change of Vaporization for CCl₄
34:19
Example V
35:41
Part A: Calculate ∆S of Transformation
37:36
Part B: Calculate ∆S of Transformation
39:10
Entropy Example Problems II

56m 44s

Intro
0:00
Example I
0:09
Example I: Calculate ∆U
1:28
Example I: Calculate Q
3:29
Example I: Calculate Cp
4:54
Example I: Calculate ∆S
6:14
Example II
7:13
Example II: Calculate W
8:14
Example II: Calculate ∆U
8:56
Example II: Calculate Q
10:18
Example II: Calculate ∆H
11:00
Example II: Calculate ∆S
12:36
Example III
18:47
Example III: Calculate ∆H
19:38
Example III: Calculate Q
21:14
Example III: Calculate ∆U
21:44
Example III: Calculate W
23:59
Example III: Calculate ∆S
24:55
Example IV
27:57
Example IV: Diagram
29:32
Example IV: Calculate W
32:27
Example IV: Calculate ∆U
36:36
Example IV: Calculate Q
38:32
Example IV: Calculate ∆H
39:00
Example IV: Calculate ∆S
40:27
Example IV: Summary
43:41
Example V
48:25
Example V: Diagram
49:05
Example V: Calculate W
50:58
Example V: Calculate ∆U
53:29
Example V: Calculate Q
53:44
Example V: Calculate ∆H
54:34
Example V: Calculate ∆S
55:01
Entropy Example Problems III

57m 6s

Intro
0:00
Example I: Isothermal Expansion
0:09
Example I: Calculate W
1:19
Example I: Calculate ∆U
1:48
Example I: Calculate Q
2:06
Example I: Calculate ∆H
2:26
Example I: Calculate ∆S
3:02
Example II: Adiabatic and Reversible Expansion
6:10
Example II: Calculate Q
6:48
Example II: Basic Equation for the Reversible Adiabatic Expansion of an Ideal Gas
8:12
Example II: Finding Volume
12:40
Example II: Finding Temperature
17:58
Example II: Calculate ∆U
19:53
Example II: Calculate W
20:59
Example II: Calculate ∆H
21:42
Example II: Calculate ∆S
23:42
Example III: Calculate the Entropy of Water Vapor
25:20
Example IV: Calculate the Molar ∆S for the Transformation
34:32
Example V
44:19
Part A: Calculate the Standard Entropy of Liquid Lead at 525°C
46:17
Part B: Calculate ∆H for the Transformation of Solid Lead from 25°C to Liquid Lead at 525°C
52:23
VI. Entropy and Probability
Entropy & Probability I

54m 35s

Intro
0:00
Entropy & Probability
0:11
Structural Model
3:05
Recall the Fundamental Equation of Thermodynamics
9:11
Two Independent Ways of Affecting the Entropy of a System
10:05
Boltzmann Definition
12:10
Omega
16:24
Definition of Omega
16:25
Energy Distribution
19:43
The Energy Distribution
19:44
In How Many Ways can N Particles be Distributed According to the Energy Distribution
23:05
Example I: In How Many Ways can the Following Distribution be Achieved
32:51
Example II: In How Many Ways can the Following Distribution be Achieved
33:51
Example III: In How Many Ways can the Following Distribution be Achieved
34:45
Example IV: In How Many Ways can the Following Distribution be Achieved
38:50
Entropy & Probability, cont.
40:57
More on Distribution
40:58
Example I Summary
41:43
Example II Summary
42:12
Distribution that Maximizes Omega
42:26
If Omega is Large, then S is Large
44:22
Two Constraints for a System to Achieve the Highest Entropy Possible
47:07
What Happened When the Energy of a System is Increased?
49:00
Entropy & Probability II

35m 5s

Intro
0:00
Volume Distribution
0:08
Distributing 2 Balls in 3 Spaces
1:43
Distributing 2 Balls in 4 Spaces
3:44
Distributing 3 Balls in 10 Spaces
5:30
Number of Ways to Distribute P Particles over N Spaces
6:05
When N is Much Larger than the Number of Particles P
7:56
Energy Distribution
25:04
Volume Distribution
25:58
Entropy, Total Entropy, & Total Omega Equations
27:34
Entropy, Total Entropy, & Total Omega Equations
27:35
VII. Spontaneity, Equilibrium, and the Fundamental Equations
Spontaneity & Equilibrium I

28m 42s

Intro
0:00
Reversible & Irreversible
0:24
Reversible vs. Irreversible
0:58
Defining Equation for Equilibrium
2:11
Defining Equation for Irreversibility (Spontaneity)
3:11
TdS ≥ dQ
5:15
Transformation in an Isolated System
11:22
Transformation in an Isolated System
11:29
Transformation at Constant Temperature
14:50
Transformation at Constant Temperature
14:51
Helmholtz Free Energy
17:26
Define: A = U - TS
17:27
Spontaneous Isothermal Process & Helmholtz Energy
20:20
Pressure-volume Work
22:02
Spontaneity & Equilibrium II

34m 38s

Intro
0:00
Transformation under Constant Temperature & Pressure
0:08
Transformation under Constant Temperature & Pressure
0:36
Define: G = U + PV - TS
3:32
Gibbs Energy
5:14
What Does This Say?
6:44
Spontaneous Process & a Decrease in G
14:12
Computing ∆G
18:54
Summary of Conditions
21:32
Constraint & Condition for Spontaneity
21:36
Constraint & Condition for Equilibrium
24:54
A Few Words About the Word Spontaneous
26:24
Spontaneous Does Not Mean Fast
26:25
Putting Hydrogen & Oxygen Together in a Flask
26:59
Spontaneous Vs. Not Spontaneous
28:14
Thermodynamically Favorable
29:03
Example: Making a Process Thermodynamically Favorable
29:34
Driving Forces for Spontaneity
31:35
Equation: ∆G = ∆H - T∆S
31:36
Always Spontaneous Process
32:39
Never Spontaneous Process
33:06
A Process That is Endothermic Can Still be Spontaneous
34:00
The Fundamental Equations of Thermodynamics

30m 50s

Intro
0:00
The Fundamental Equations of Thermodynamics
0:44
Mechanical Properties of a System
0:45
Fundamental Properties of a System
1:16
Composite Properties of a System
1:44
General Condition of Equilibrium
3:16
Composite Functions & Their Differentiations
6:11
dH = TdS + VdP
7:53
dA = -SdT - PdV
9:26
dG = -SdT + VdP
10:22
Summary of Equations
12:10
Equation #1
14:33
Equation #2
15:15
Equation #3
15:58
Equation #4
16:42
Maxwell's Relations
20:20
Maxwell's Relations
20:21
Isothermal Volume-Dependence of Entropy & Isothermal Pressure-Dependence of Entropy
26:21
The General Thermodynamic Equations of State

34m 6s

Intro
0:00
The General Thermodynamic Equations of State
0:10
Equations of State for Liquids & Solids
0:52
More General Condition for Equilibrium
4:02
General Conditions: Equation that Relates P to Functions of T & V
6:20
The Second Fundamental Equation of Thermodynamics
11:10
Equation 1
17:34
Equation 2
21:58
Recall the General Expression for Cp - Cv
28:11
For the Joule-Thomson Coefficient
30:44
Joule-Thomson Inversion Temperature
32:12
Properties of the Helmholtz & Gibbs Energies

39m 18s

Intro
0:00
Properties of the Helmholtz & Gibbs Energies
0:10
Equating the Differential Coefficients
1:34
An Increase in T; a Decrease in A
3:25
An Increase in V; a Decrease in A
6:04
We Do the Same Thing for G
8:33
Increase in T; Decrease in G
10:50
Increase in P; Decrease in G
11:36
Gibbs Energy of a Pure Substance at a Constant Temperature from 1 atm to any Other Pressure.
14:12
If the Substance is a Liquid or a Solid, then Volume can be Treated as a Constant
18:57
For an Ideal Gas
22:18
Special Note
24:56
Temperature Dependence of Gibbs Energy
27:02
Temperature Dependence of Gibbs Energy #1
27:52
Temperature Dependence of Gibbs Energy #2
29:01
Temperature Dependence of Gibbs Energy #3
29:50
Temperature Dependence of Gibbs Energy #4
34:50
The Entropy of the Universe & the Surroundings

19m 40s

Intro
0:00
Entropy of the Universe & the Surroundings
0:08
Equation: ∆G = ∆H - T∆S
0:20
Conditions of Constant Temperature & Pressure
1:14
Reversible Process
3:14
Spontaneous Process & the Entropy of the Universe
5:20
Tips for Remembering Everything
12:40
Verify Using Known Spontaneous Process
14:51
VIII. Free Energy Example Problems
Free Energy Example Problems I

54m 16s

Intro
0:00
Example I
0:11
Example I: Deriving a Function for Entropy (S)
2:06
Example I: Deriving a Function for V
5:55
Example I: Deriving a Function for H
8:06
Example I: Deriving a Function for U
12:06
Example II
15:18
Example III
21:52
Example IV
26:12
Example IV: Part A
26:55
Example IV: Part B
28:30
Example IV: Part C
30:25
Example V
33:45
Example VI
40:46
Example VII
43:43
Example VII: Part A
44:46
Example VII: Part B
50:52
Example VII: Part C
51:56
Free Energy Example Problems II

31m 17s

Intro
0:00
Example I
0:09
Example II
5:18
Example III
8:22
Example IV
12:32
Example V
17:14
Example VI
20:34
Example VI: Part A
21:04
Example VI: Part B
23:56
Example VI: Part C
27:56
Free Energy Example Problems III

45m

Intro
0:00
Example I
0:10
Example II
15:03
Example III
21:47
Example IV
28:37
Example IV: Part A
29:33
Example IV: Part B
36:09
Example IV: Part C
40:34
Three Miscellaneous Example Problems

58m 5s

Intro
0:00
Example I
0:41
Part A: Calculating ∆H
3:55
Part B: Calculating ∆S
15:13
Example II
24:39
Part A: Final Temperature of the System
26:25
Part B: Calculating ∆S
36:57
Example III
46:49
IX. Equation Review for Thermodynamics
Looking Back Over Everything: All the Equations in One Place

25m 20s

Intro
0:00
Work, Heat, and Energy
0:18
Definition of Work, Energy, Enthalpy, and Heat Capacities
0:23
Heat Capacities for an Ideal Gas
3:40
Path Property & State Property
3:56
Energy Differential
5:04
Enthalpy Differential
5:40
Joule's Law & Joule-Thomson Coefficient
6:23
Coefficient of Thermal Expansion & Coefficient of Compressibility
7:01
Enthalpy of a Substance at Any Other Temperature
7:29
Enthalpy of a Reaction at Any Other Temperature
8:01
Entropy
8:53
Definition of Entropy
8:54
Clausius Inequality
9:11
Entropy Changes in Isothermal Systems
9:44
The Fundamental Equation of Thermodynamics
10:12
Expressing Entropy Changes in Terms of Properties of the System
10:42
Entropy Changes in the Ideal Gas
11:22
Third Law Entropies
11:38
Entropy Changes in Chemical Reactions
14:02
Statistical Definition of Entropy
14:34
Omega for the Spatial & Energy Distribution
14:47
Spontaneity and Equilibrium
15:43
Helmholtz Energy & Gibbs Energy
15:44
Condition for Spontaneity & Equilibrium
16:24
Condition for Spontaneity with Respect to Entropy
17:58
The Fundamental Equations
18:30
Maxwell's Relations
19:04
The Thermodynamic Equations of State
20:07
Energy & Enthalpy Differentials
21:08
Joule's Law & Joule-Thomson Coefficient
21:59
Relationship Between Constant Pressure & Constant Volume Heat Capacities
23:14
One Final Equation - Just for Fun
24:04
X. Quantum Mechanics Preliminaries
Complex Numbers

34m 25s

Intro
0:00
Complex Numbers
0:11
Representing Complex Numbers in the 2-Dimmensional Plane
0:56
Addition of Complex Numbers
2:35
Subtraction of Complex Numbers
3:17
Multiplication of Complex Numbers
3:47
Division of Complex Numbers
6:04
r & θ
8:04
Euler's Formula
11:00
Polar Exponential Representation of the Complex Numbers
11:22
Example I
14:25
Example II
15:21
Example III
16:58
Example IV
18:35
Example V
20:40
Example VI
21:32
Example VII
25:22
Probability & Statistics

59m 57s

Intro
0:00
Probability & Statistics
1:51
Normalization Condition
1:52
Define the Mean or Average of x
11:04
Example I: Calculate the Mean of x
14:57
Example II: Calculate the Second Moment of the Data in Example I
22:39
Define the Second Central Moment or Variance
25:26
Define the Second Central Moment or Variance
25:27
1st Term
32:16
2nd Term
32:40
3rd Term
34:07
Continuous Distributions
35:47
Continuous Distributions
35:48
Probability Density
39:30
Probability Density
39:31
Normalization Condition
46:51
Example III
50:13
Part A - Show that P(x) is Normalized
51:40
Part B - Calculate the Average Position of the Particle Along the Interval
54:31
Important Things to Remember
58:24
SchrÓ§dinger Equation & Operators

42m 5s

Intro
0:00
Schrӧdinger Equation & Operators
0:16
Relation Between a Photon's Momentum & Its Wavelength
0:17
Louis de Broglie: Wavelength for Matter
0:39
Schrӧdinger Equation
1:19
Definition of Ψ(x)
3:31
Quantum Mechanics
5:02
Operators
7:51
Example I
10:10
Example II
11:53
Example III
14:24
Example IV
17:35
Example V
19:59
Example VI
22:39
Operators Can Be Linear or Non Linear
27:58
Operators Can Be Linear or Non Linear
28:34
Example VII
32:47
Example VIII
36:55
Example IX
39:29
SchrÓ§dinger Equation as an Eigenvalue Problem

30m 26s

Intro
0:00
Schrӧdinger Equation as an Eigenvalue Problem
0:10
Operator: Multiplying the Original Function by Some Scalar
0:11
Operator, Eigenfunction, & Eigenvalue
4:42
Example: Eigenvalue Problem
8:00
Schrӧdinger Equation as an Eigenvalue Problem
9:24
Hamiltonian Operator
15:09
Quantum Mechanical Operators
16:46
Kinetic Energy Operator
19:16
Potential Energy Operator
20:02
Total Energy Operator
21:12
Classical Point of View
21:48
Linear Momentum Operator
24:02
Example I
26:01
The Plausibility of the SchrÓ§dinger Equation

21m 34s

Intro
0:00
The Plausibility of the Schrӧdinger Equation
1:16
The Plausibility of the Schrӧdinger Equation, Part 1
1:17
The Plausibility of the Schrӧdinger Equation, Part 2
8:24
The Plausibility of the Schrӧdinger Equation, Part 3
13:45
XI. The Particle in a Box
The Particle in a Box Part I

56m 22s

Intro
0:00
Free Particle in a Box
0:28
Definition of a Free Particle in a Box
0:29
Amplitude of the Matter Wave
6:22
Intensity of the Wave
6:53
Probability Density
9:39
Probability that the Particle is Located Between x & dx
10:54
Probability that the Particle will be Found Between o & a
12:35
Wave Function & the Particle
14:59
Boundary Conditions
19:22
What Happened When There is No Constraint on the Particle
27:54
Diagrams
34:12
More on Probability Density
40:53
The Correspondence Principle
46:45
The Correspondence Principle
46:46
Normalizing the Wave Function
47:46
Normalizing the Wave Function
47:47
Normalized Wave Function & Normalization Constant
52:24
The Particle in a Box Part II

45m 24s

Intro
0:00
Free Particle in a Box
0:08
Free Particle in a 1-dimensional Box
0:09
For a Particle in a Box
3:57
Calculating Average Values & Standard Deviations
5:42
Average Value for the Position of a Particle
6:32
Standard Deviations for the Position of a Particle
10:51
Recall: Energy & Momentum are Represented by Operators
13:33
Recall: Schrӧdinger Equation in Operator Form
15:57
Average Value of a Physical Quantity that is Associated with an Operator
18:16
Average Momentum of a Free Particle in a Box
20:48
The Uncertainty Principle
24:42
Finding the Standard Deviation of the Momentum
25:08
Expression for the Uncertainty Principle
35:02
Summary of the Uncertainty Principle
41:28
The Particle in a Box Part III

48m 43s

Intro
0:00
2-Dimension
0:12
Dimension 2
0:31
Boundary Conditions
1:52
Partial Derivatives
4:27
Example I
6:08
The Particle in a Box, cont.
11:28
Operator Notation
12:04
Symbol for the Laplacian
13:50
The Equation Becomes…
14:30
Boundary Conditions
14:54
Separation of Variables
15:33
Solution to the 1-dimensional Case
16:31
Normalization Constant
22:32
3-Dimension
28:30
Particle in a 3-dimensional Box
28:31
In Del Notation
32:22
The Solutions
34:51
Expressing the State of the System for a Particle in a 3D Box
39:10
Energy Level & Degeneracy
43:35
XII. Postulates and Principles of Quantum Mechanics
The Postulates & Principles of Quantum Mechanics, Part I

46m 18s

Intro
0:00
Postulate I
0:31
Probability That The Particle Will Be Found in a Differential Volume Element
0:32
Example I: Normalize This Wave Function
11:30
Postulate II
18:20
Postulate II
18:21
Quantum Mechanical Operators: Position
20:48
Quantum Mechanical Operators: Kinetic Energy
21:57
Quantum Mechanical Operators: Potential Energy
22:42
Quantum Mechanical Operators: Total Energy
22:57
Quantum Mechanical Operators: Momentum
23:22
Quantum Mechanical Operators: Angular Momentum
23:48
More On The Kinetic Energy Operator
24:48
Angular Momentum
28:08
Angular Momentum Overview
28:09
Angular Momentum Operator in Quantum Mechanic
31:34
The Classical Mechanical Observable
32:56
Quantum Mechanical Operator
37:01
Getting the Quantum Mechanical Operator from the Classical Mechanical Observable
40:16
Postulate II, cont.
43:40
Quantum Mechanical Operators are Both Linear & Hermetical
43:41
The Postulates & Principles of Quantum Mechanics, Part II

39m 28s

Intro
0:00
Postulate III
0:09
Postulate III: Part I
0:10
Postulate III: Part II
5:56
Postulate III: Part III
12:43
Postulate III: Part IV
18:28
Postulate IV
23:57
Postulate IV
23:58
Postulate V
27:02
Postulate V
27:03
Average Value
36:38
Average Value
36:39
The Postulates & Principles of Quantum Mechanics, Part III

35m 32s

Intro
0:00
The Postulates & Principles of Quantum Mechanics, Part III
0:10
Equations: Linear & Hermitian
0:11
Introduction to Hermitian Property
3:36
Eigenfunctions are Orthogonal
9:55
The Sequence of Wave Functions for the Particle in a Box forms an Orthonormal Set
14:34
Definition of Orthogonality
16:42
Definition of Hermiticity
17:26
Hermiticity: The Left Integral
23:04
Hermiticity: The Right Integral
28:47
Hermiticity: Summary
34:06
The Postulates & Principles of Quantum Mechanics, Part IV

29m 55s

Intro
0:00
The Postulates & Principles of Quantum Mechanics, Part IV
0:09
Operators can be Applied Sequentially
0:10
Sample Calculation 1
2:41
Sample Calculation 2
5:18
Commutator of Two Operators
8:16
The Uncertainty Principle
19:01
In the Case of Linear Momentum and Position Operator
23:14
When the Commutator of Two Operators Equals to Zero
26:31
XIII. Postulates and Principles Example Problems, Including Particle in a Box
Example Problems I

54m 25s

Intro
0:00
Example I: Three Dimensional Box & Eigenfunction of The Laplacian Operator
0:37
Example II: Positions of a Particle in a 1-dimensional Box
15:46
Example III: Transition State & Frequency
29:29
Example IV: Finding a Particle in a 1-dimensional Box
35:03
Example V: Degeneracy & Energy Levels of a Particle in a Box
44:59
Example Problems II

46m 58s

Intro
0:00
Review
0:25
Wave Function
0:26
Normalization Condition
2:28
Observable in Classical Mechanics & Linear/Hermitian Operator in Quantum Mechanics
3:36
Hermitian
6:11
Eigenfunctions & Eigenvalue
8:20
Normalized Wave Functions
12:00
Average Value
13:42
If Ψ is Written as a Linear Combination
15:44
Commutator
16:45
Example I: Normalize The Wave Function
19:18
Example II: Probability of Finding of a Particle
22:27
Example III: Orthogonal
26:00
Example IV: Average Value of the Kinetic Energy Operator
30:22
Example V: Evaluate These Commutators
39:02
Example Problems III

44m 11s

Intro
0:00
Example I: Good Candidate for a Wave Function
0:08
Example II: Variance of the Energy
7:00
Example III: Evaluate the Angular Momentum Operators
15:00
Example IV: Real Eigenvalues Imposes the Hermitian Property on Operators
28:44
Example V: A Demonstration of Why the Eigenfunctions of Hermitian Operators are Orthogonal
35:33
XIV. The Harmonic Oscillator
The Harmonic Oscillator I

35m 33s

Intro
0:00
The Harmonic Oscillator
0:10
Harmonic Motion
0:11
Classical Harmonic Oscillator
4:38
Hooke's Law
8:18
Classical Harmonic Oscillator, cont.
10:33
General Solution for the Differential Equation
15:16
Initial Position & Velocity
16:05
Period & Amplitude
20:42
Potential Energy of the Harmonic Oscillator
23:20
Kinetic Energy of the Harmonic Oscillator
26:37
Total Energy of the Harmonic Oscillator
27:23
Conservative System
34:37
The Harmonic Oscillator II

43m 4s

Intro
0:00
The Harmonic Oscillator II
0:08
Diatomic Molecule
0:10
Notion of Reduced Mass
5:27
Harmonic Oscillator Potential & The Intermolecular Potential of a Vibrating Molecule
7:33
The Schrӧdinger Equation for the 1-dimensional Quantum Mechanic Oscillator
14:14
Quantized Values for the Energy Level
15:46
Ground State & the Zero-Point Energy
21:50
Vibrational Energy Levels
25:18
Transition from One Energy Level to the Next
26:42
Fundamental Vibrational Frequency for Diatomic Molecule
34:57
Example: Calculate k
38:01
The Harmonic Oscillator III

26m 30s

Intro
0:00
The Harmonic Oscillator III
0:09
The Wave Functions Corresponding to the Energies
0:10
Normalization Constant
2:34
Hermite Polynomials
3:22
First Few Hermite Polynomials
4:56
First Few Wave-Functions
6:37
Plotting the Probability Density of the Wave-Functions
8:37
Probability Density for Large Values of r
14:24
Recall: Odd Function & Even Function
19:05
More on the Hermite Polynomials
20:07
Recall: If f(x) is Odd
20:36
Average Value of x
22:31
Average Value of Momentum
23:56
XV. The Rigid Rotator
The Rigid Rotator I

41m 10s

Intro
0:00
Possible Confusion from the Previous Discussion
0:07
Possible Confusion from the Previous Discussion
0:08
Rotation of a Single Mass Around a Fixed Center
8:17
Rotation of a Single Mass Around a Fixed Center
8:18
Angular Velocity
12:07
Rotational Inertia
13:24
Rotational Frequency
15:24
Kinetic Energy for a Linear System
16:38
Kinetic Energy for a Rotational System
17:42
Rotating Diatomic Molecule
19:40
Rotating Diatomic Molecule: Part 1
19:41
Rotating Diatomic Molecule: Part 2
24:56
Rotating Diatomic Molecule: Part 3
30:04
Hamiltonian of the Rigid Rotor
36:48
Hamiltonian of the Rigid Rotor
36:49
The Rigid Rotator II

30m 32s

Intro
0:00
The Rigid Rotator II
0:08
Cartesian Coordinates
0:09
Spherical Coordinates
1:55
r
6:15
θ
6:28
φ
7:00
Moving a Distance 'r'
8:17
Moving a Distance 'r' in the Spherical Coordinates
11:49
For a Rigid Rotator, r is Constant
13:57
Hamiltonian Operator
15:09
Square of the Angular Momentum Operator
17:34
Orientation of the Rotation in Space
19:44
Wave Functions for the Rigid Rotator
20:40
The Schrӧdinger Equation for the Quantum Mechanic Rigid Rotator
21:24
Energy Levels for the Rigid Rotator
26:58
The Rigid Rotator III

35m 19s

Intro
0:00
The Rigid Rotator III
0:11
When a Rotator is Subjected to Electromagnetic Radiation
1:24
Selection Rule
2:13
Frequencies at Which Absorption Transitions Occur
6:24
Energy Absorption & Transition
10:54
Energy of the Individual Levels Overview
20:58
Energy of the Individual Levels: Diagram
23:45
Frequency Required to Go from J to J + 1
25:53
Using Separation Between Lines on the Spectrum to Calculate Bond Length
28:02
Example I: Calculating Rotational Inertia & Bond Length
29:18
Example I: Calculating Rotational Inertia
29:19
Example I: Calculating Bond Length
32:56
XVI. Oscillator and Rotator Example Problems
Example Problems I

33m 48s

Intro
0:00
Equations Review
0:11
Energy of the Harmonic Oscillator
0:12
Selection Rule
3:02
Observed Frequency of Radiation
3:27
Harmonic Oscillator Wave Functions
5:52
Rigid Rotator
7:26
Selection Rule for Rigid Rotator
9:15
Frequency of Absorption
9:35
Wave Numbers
10:58
Example I: Calculate the Reduced Mass of the Hydrogen Atom
11:44
Example II: Calculate the Fundamental Vibration Frequency & the Zero-Point Energy of This Molecule
13:37
Example III: Show That the Product of Two Even Functions is even
19:35
Example IV: Harmonic Oscillator
24:56
Example Problems II

46m 43s

Intro
0:00
Example I: Harmonic Oscillator
0:12
Example II: Harmonic Oscillator
23:26
Example III: Calculate the RMS Displacement of the Molecules
38:12
XVII. The Hydrogen Atom
The Hydrogen Atom I

40m

Intro
0:00
The Hydrogen Atom I
1:31
Review of the Rigid Rotator
1:32
Hydrogen Atom & the Coulomb Potential
2:50
Using the Spherical Coordinates
6:33
Applying This Last Expression to Equation 1
10:19
Angular Component & Radial Component
13:26
Angular Equation
15:56
Solution for F(φ)
19:32
Determine The Normalization Constant
20:33
Differential Equation for T(a)
24:44
Legendre Equation
27:20
Legendre Polynomials
31:20
The Legendre Polynomials are Mutually Orthogonal
35:40
Limits
37:17
Coefficients
38:28
The Hydrogen Atom II

35m 58s

Intro
0:00
Associated Legendre Functions
0:07
Associated Legendre Functions
0:08
First Few Associated Legendre Functions
6:39
s, p, & d Orbital
13:24
The Normalization Condition
15:44
Spherical Harmonics
20:03
Equations We Have Found
20:04
Wave Functions for the Angular Component & Rigid Rotator
24:36
Spherical Harmonics Examples
25:40
Angular Momentum
30:09
Angular Momentum
30:10
Square of the Angular Momentum
35:38
Energies of the Rigid Rotator
38:21
The Hydrogen Atom III

36m 18s

Intro
0:00
The Hydrogen Atom III
0:34
Angular Momentum is a Vector Quantity
0:35
The Operators Corresponding to the Three Components of Angular Momentum Operator: In Cartesian Coordinates
1:30
The Operators Corresponding to the Three Components of Angular Momentum Operator: In Spherical Coordinates
3:27
Z Component of the Angular Momentum Operator & the Spherical Harmonic
5:28
Magnitude of the Angular Momentum Vector
20:10
Classical Interpretation of Angular Momentum
25:22
Projection of the Angular Momentum Vector onto the xy-plane
33:24
The Hydrogen Atom IV

33m 55s

Intro
0:00
The Hydrogen Atom IV
0:09
The Equation to Find R( r )
0:10
Relation Between n & l
3:50
The Solutions for the Radial Functions
5:08
Associated Laguerre Polynomials
7:58
1st Few Associated Laguerre Polynomials
8:55
Complete Wave Function for the Atomic Orbitals of the Hydrogen Atom
12:24
The Normalization Condition
15:06
In Cartesian Coordinates
18:10
Working in Polar Coordinates
20:48
Principal Quantum Number
21:58
Angular Momentum Quantum Number
22:35
Magnetic Quantum Number
25:55
Zeeman Effect
30:45
The Hydrogen Atom V: Where We Are

51m 53s

Intro
0:00
The Hydrogen Atom V: Where We Are
0:13
Review
0:14
Let's Write Out ψ₂₁₁
7:32
Angular Momentum of the Electron
14:52
Representation of the Wave Function
19:36
Radial Component
28:02
Example: 1s Orbital
28:34
Probability for Radial Function
33:46
1s Orbital: Plotting Probability Densities vs. r
35:47
2s Orbital: Plotting Probability Densities vs. r
37:46
3s Orbital: Plotting Probability Densities vs. r
38:49
4s Orbital: Plotting Probability Densities vs. r
39:34
2p Orbital: Plotting Probability Densities vs. r
40:12
3p Orbital: Plotting Probability Densities vs. r
41:02
4p Orbital: Plotting Probability Densities vs. r
41:51
3d Orbital: Plotting Probability Densities vs. r
43:18
4d Orbital: Plotting Probability Densities vs. r
43:48
Example I: Probability of Finding an Electron in the 2s Orbital of the Hydrogen
45:40
The Hydrogen Atom VI

51m 53s

Intro
0:00
The Hydrogen Atom VI
0:07
Last Lesson Review
0:08
Spherical Component
1:09
Normalization Condition
2:02
Complete 1s Orbital Wave Function
4:08
1s Orbital Wave Function
4:09
Normalization Condition
6:28
Spherically Symmetric
16:00
Average Value
17:52
Example I: Calculate the Region of Highest Probability for Finding the Electron
21:19
2s Orbital Wave Function
25:32
2s Orbital Wave Function
25:33
Average Value
28:56
General Formula
32:24
The Hydrogen Atom VII

34m 29s

Intro
0:00
The Hydrogen Atom VII
0:12
p Orbitals
1:30
Not Spherically Symmetric
5:10
Recall That the Spherical Harmonics are Eigenfunctions of the Hamiltonian Operator
6:50
Any Linear Combination of These Orbitals Also Has The Same Energy
9:16
Functions of Real Variables
15:53
Solving for Px
16:50
Real Spherical Harmonics
21:56
Number of Nodes
32:56
XVIII. Hydrogen Atom Example Problems
Hydrogen Atom Example Problems I

43m 49s

Intro
0:00
Example I: Angular Momentum & Spherical Harmonics
0:20
Example II: Pair-wise Orthogonal Legendre Polynomials
16:40
Example III: General Normalization Condition for the Legendre Polynomials
25:06
Example IV: Associated Legendre Functions
32:13
The Hydrogen Atom Example Problems II

1h 1m 57s

Intro
0:00
Example I: Normalization & Pair-wise Orthogonal
0:13
Part 1: Normalized
0:43
Part 2: Pair-wise Orthogonal
16:53
Example II: Show Explicitly That the Following Statement is True for Any Integer n
27:10
Example III: Spherical Harmonics
29:26
Angular Momentum Cones
56:37
Angular Momentum Cones
56:38
Physical Interpretation of Orbital Angular Momentum in Quantum mechanics
1:00:16
The Hydrogen Atom Example Problems III

48m 33s

Intro
0:00
Example I: Show That ψ₂₁₁ is Normalized
0:07
Example II: Show That ψ₂₁₁ is Orthogonal to ψ₃₁₀
11:48
Example III: Probability That a 1s Electron Will Be Found Within 1 Bohr Radius of The Nucleus
18:35
Example IV: Radius of a Sphere
26:06
Example V: Calculate <r> for the 2s Orbital of the Hydrogen-like Atom
36:33
The Hydrogen Atom Example Problems IV

48m 33s

Intro
0:00
Example I: Probability Density vs. Radius Plot
0:11
Example II: Hydrogen Atom & The Coulombic Potential
14:16
Example III: Find a Relation Among <K>, <V>, & <E>
25:47
Example IV: Quantum Mechanical Virial Theorem
48:32
Example V: Find the Variance for the 2s Orbital
54:13
The Hydrogen Atom Example Problems V

48m 33s

Intro
0:00
Example I: Derive a Formula for the Degeneracy of a Given Level n
0:11
Example II: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ
8:30
Example III: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ
23:01
Example IV: Orbital Functions
31:51
XIX. Spin Quantum Number and Atomic Term Symbols
Spin Quantum Number: Term Symbols I

59m 18s

Intro
0:00
Quantum Numbers Specify an Orbital
0:24
n
1:10
l
1:20
m
1:35
4th Quantum Number: s
2:02
Spin Orbitals
7:03
Spin Orbitals
7:04
Multi-electron Atoms
11:08
Term Symbols
18:08
Russell-Saunders Coupling & The Atomic Term Symbol
18:09
Example: Configuration for C
27:50
Configuration for C: 1s²2s²2p²
27:51
Drawing Every Possible Arrangement
31:15
Term Symbols
45:24
Microstate
50:54
Spin Quantum Number: Term Symbols II

34m 54s

Intro
0:00
Microstates
0:25
We Started With 21 Possible Microstates
0:26
³P State
2:05
Microstates in ³P Level
5:10
¹D State
13:16
³P State
16:10
²P₂ State
17:34
³P₁ State
18:34
³P₀ State
19:12
9 Microstates in ³P are Subdivided
19:40
¹S State
21:44
Quicker Way to Find the Different Values of J for a Given Basic Term Symbol
22:22
Ground State
26:27
Hund's Empirical Rules for Specifying the Term Symbol for the Ground Electronic State
27:29
Hund's Empirical Rules: 1
28:24
Hund's Empirical Rules: 2
29:22
Hund's Empirical Rules: 3 - Part A
30:22
Hund's Empirical Rules: 3 - Part B
31:18
Example: 1s²2s²2p²
31:54
Spin Quantum Number: Term Symbols III

38m 3s

Intro
0:00
Spin Quantum Number: Term Symbols III
0:14
Deriving the Term Symbols for the p² Configuration
0:15
Table: MS vs. ML
3:57
¹D State
16:21
³P State
21:13
¹S State
24:48
J Value
25:32
Degeneracy of the Level
27:28
When Given r Electrons to Assign to n Equivalent Spin Orbitals
30:18
p² Configuration
32:51
Complementary Configurations
35:12
Term Symbols & Atomic Spectra

57m 49s

Intro
0:00
Lyman Series
0:09
Spectroscopic Term Symbols
0:10
Lyman Series
3:04
Hydrogen Levels
8:21
Hydrogen Levels
8:22
Term Symbols & Atomic Spectra
14:17
Spin-Orbit Coupling
14:18
Selection Rules for Atomic Spectra
21:31
Selection Rules for Possible Transitions
23:56
Wave Numbers for The Transitions
28:04
Example I: Calculate the Frequencies of the Allowed Transitions from (4d) ²D →(2p) ²P
32:23
Helium Levels
49:50
Energy Levels for Helium
49:51
Transitions & Spin Multiplicity
52:27
Transitions & Spin Multiplicity
52:28
XX. Term Symbols Example Problems
Example Problems I

1h 1m 20s

Intro
0:00
Example I: What are the Term Symbols for the np¹ Configuration?
0:10
Example II: What are the Term Symbols for the np² Configuration?
20:38
Example III: What are the Term Symbols for the np³ Configuration?
40:46
Example Problems II

56m 34s

Intro
0:00
Example I: Find the Term Symbols for the nd² Configuration
0:11
Example II: Find the Term Symbols for the 1s¹2p¹ Configuration
27:02
Example III: Calculate the Separation Between the Doublets in the Lyman Series for Atomic Hydrogen
41:41
Example IV: Calculate the Frequencies of the Lines for the (4d) ²D → (3p) ²P Transition
48:53
XXI. Equation Review for Quantum Mechanics
Quantum Mechanics: All the Equations in One Place

18m 24s

Intro
0:00
Quantum Mechanics Equations
0:37
De Broglie Relation
0:38
Statistical Relations
1:00
The Schrӧdinger Equation
1:50
The Particle in a 1-Dimensional Box of Length a
3:09
The Particle in a 2-Dimensional Box of Area a x b
3:48
The Particle in a 3-Dimensional Box of Area a x b x c
4:22
The Schrӧdinger Equation Postulates
4:51
The Normalization Condition
5:40
The Probability Density
6:51
Linear
7:47
Hermitian
8:31
Eigenvalues & Eigenfunctions
8:55
The Average Value
9:29
Eigenfunctions of Quantum Mechanics Operators are Orthogonal
10:53
Commutator of Two Operators
10:56
The Uncertainty Principle
11:41
The Harmonic Oscillator
13:18
The Rigid Rotator
13:52
Energy of the Hydrogen Atom
14:30
Wavefunctions, Radial Component, and Associated Laguerre Polynomial
14:44
Angular Component or Spherical Harmonic
15:16
Associated Legendre Function
15:31
Principal Quantum Number
15:43
Angular Momentum Quantum Number
15:50
Magnetic Quantum Number
16:21
z-component of the Angular Momentum of the Electron
16:53
Atomic Spectroscopy: Term Symbols
17:14
Atomic Spectroscopy: Selection Rules
18:03
XXII. Molecular Spectroscopy
Spectroscopic Overview: Which Equation Do I Use & Why

50m 2s

Intro
0:00
Spectroscopic Overview: Which Equation Do I Use & Why
1:02
Lesson Overview
1:03
Rotational & Vibrational Spectroscopy
4:01
Frequency of Absorption/Emission
6:04
Wavenumbers in Spectroscopy
8:10
Starting State vs. Excited State
10:10
Total Energy of a Molecule (Leaving out the Electronic Energy)
14:02
Energy of Rotation: Rigid Rotor
15:55
Energy of Vibration: Harmonic Oscillator
19:08
Equation of the Spectral Lines
23:22
Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)
28:37
Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)
28:38
Vibration-Rotation Interaction
33:46
Centrifugal Distortion
36:27
Anharmonicity
38:28
Correcting for All Three Simultaneously
41:03
Spectroscopic Parameters
44:26
Summary
47:32
Harmonic Oscillator-Rigid Rotor Approximation
47:33
Vibration-Rotation Interaction
48:14
Centrifugal Distortion
48:20
Anharmonicity
48:28
Correcting for All Three Simultaneously
48:44
Vibration-Rotation

59m 47s

Intro
0:00
Vibration-Rotation
0:37
What is Molecular Spectroscopy?
0:38
Microwave, Infrared Radiation, Visible & Ultraviolet
1:53
Equation for the Frequency of the Absorbed Radiation
4:54
Wavenumbers
6:15
Diatomic Molecules: Energy of the Harmonic Oscillator
8:32
Selection Rules for Vibrational Transitions
10:35
Energy of the Rigid Rotator
16:29
Angular Momentum of the Rotator
21:38
Rotational Term F(J)
26:30
Selection Rules for Rotational Transition
29:30
Vibration Level & Rotational States
33:20
Selection Rules for Vibration-Rotation
37:42
Frequency of Absorption
39:32
Diagram: Energy Transition
45:55
Vibration-Rotation Spectrum: HCl
51:27
Vibration-Rotation Spectrum: Carbon Monoxide
54:30
Vibration-Rotation Interaction

46m 22s

Intro
0:00
Vibration-Rotation Interaction
0:13
Vibration-Rotation Spectrum: HCl
0:14
Bond Length & Vibrational State
4:23
Vibration Rotation Interaction
10:18
Case 1
12:06
Case 2
17:17
Example I: HCl Vibration-Rotation Spectrum
22:58
Rotational Constant for the 0 & 1 Vibrational State
26:30
Equilibrium Bond Length for the 1 Vibrational State
39:42
Equilibrium Bond Length for the 0 Vibrational State
42:13
Bₑ & αₑ
44:54
The Non-Rigid Rotator

29m 24s

Intro
0:00
The Non-Rigid Rotator
0:09
Pure Rotational Spectrum
0:54
The Selection Rules for Rotation
3:09
Spacing in the Spectrum
5:04
Centrifugal Distortion Constant
9:00
Fundamental Vibration Frequency
11:46
Observed Frequencies of Absorption
14:14
Difference between the Rigid Rotator & the Adjusted Rigid Rotator
16:51
Adjusted Rigid Rotator
21:31
Observed Frequencies of Absorption
26:26
The Anharmonic Oscillator

30m 53s

Intro
0:00
The Anharmonic Oscillator
0:09
Vibration-Rotation Interaction & Centrifugal Distortion
0:10
Making Corrections to the Harmonic Oscillator
4:50
Selection Rule for the Harmonic Oscillator
7:50
Overtones
8:40
True Oscillator
11:46
Harmonic Oscillator Energies
13:16
Anharmonic Oscillator Energies
13:33
Observed Frequencies of the Overtones
15:09
True Potential
17:22
HCl Vibrational Frequencies: Fundamental & First Few Overtones
21:10
Example I: Vibrational States & Overtones of the Vibrational Spectrum
22:42
Example I: Part A - First 4 Vibrational States
23:44
Example I: Part B - Fundamental & First 3 Overtones
25:31
Important Equations
27:45
Energy of the Q State
29:14
The Difference in Energy between 2 Successive States
29:23
Difference in Energy between 2 Spectral Lines
29:40
Electronic Transitions

1h 1m 33s

Intro
0:00
Electronic Transitions
0:16
Electronic State & Transition
0:17
Total Energy of the Diatomic Molecule
3:34
Vibronic Transitions
4:30
Selection Rule for Vibronic Transitions
9:11
More on Vibronic Transitions
10:08
Frequencies in the Spectrum
16:46
Difference of the Minima of the 2 Potential Curves
24:48
Anharmonic Zero-point Vibrational Energies of the 2 States
26:24
Frequency of the 0 → 0 Vibronic Transition
27:54
Making the Equation More Compact
29:34
Spectroscopic Parameters
32:11
Franck-Condon Principle
34:32
Example I: Find the Values of the Spectroscopic Parameters for the Upper Excited State
47:27
Table of Electronic States and Parameters
56:41
XXIII. Molecular Spectroscopy Example Problems
Example Problems I

33m 47s

Intro
0:00
Example I: Calculate the Bond Length
0:10
Example II: Calculate the Rotational Constant
7:39
Example III: Calculate the Number of Rotations
10:54
Example IV: What is the Force Constant & Period of Vibration?
16:31
Example V: Part A - Calculate the Fundamental Vibration Frequency
21:42
Example V: Part B - Calculate the Energies of the First Three Vibrational Levels
24:12
Example VI: Calculate the Frequencies of the First 2 Lines of the R & P Branches of the Vib-Rot Spectrum of HBr
26:28
Example Problems II

1h 1m 5s

Intro
0:00
Example I: Calculate the Frequencies of the Transitions
0:09
Example II: Specify Which Transitions are Allowed & Calculate the Frequencies of These Transitions
22:07
Example III: Calculate the Vibrational State & Equilibrium Bond Length
34:31
Example IV: Frequencies of the Overtones
49:28
Example V: Vib-Rot Interaction, Centrifugal Distortion, & Anharmonicity
54:47
Example Problems III

33m 31s

Intro
0:00
Example I: Part A - Derive an Expression for ∆G( r )
0:10
Example I: Part B - Maximum Vibrational Quantum Number
6:10
Example II: Part A - Derive an Expression for the Dissociation Energy of the Molecule
8:29
Example II: Part B - Equation for ∆G( r )
14:00
Example III: How Many Vibrational States are There for Br₂ before the Molecule Dissociates
18:16
Example IV: Find the Difference between the Two Minima of the Potential Energy Curves
20:57
Example V: Rotational Spectrum
30:51
XXIV. Statistical Thermodynamics
Statistical Thermodynamics: The Big Picture

1h 1m 15s

Intro
0:00
Statistical Thermodynamics: The Big Picture
0:10
Our Big Picture Goal
0:11
Partition Function (Q)
2:42
The Molecular Partition Function (q)
4:00
Consider a System of N Particles
6:54
Ensemble
13:22
Energy Distribution Table
15:36
Probability of Finding a System with Energy
16:51
The Partition Function
21:10
Microstate
28:10
Entropy of the Ensemble
30:34
Entropy of the System
31:48
Expressing the Thermodynamic Functions in Terms of The Partition Function
39:21
The Partition Function
39:22
Pi & U
41:20
Entropy of the System
44:14
Helmholtz Energy
48:15
Pressure of the System
49:32
Enthalpy of the System
51:46
Gibbs Free Energy
52:56
Heat Capacity
54:30
Expressing Q in Terms of the Molecular Partition Function (q)
59:31
Indistinguishable Particles
1:02:16
N is the Number of Particles in the System
1:03:27
The Molecular Partition Function
1:05:06
Quantum States & Degeneracy
1:07:46
Thermo Property in Terms of ln Q
1:10:09
Example: Thermo Property in Terms of ln Q
1:13:23
Statistical Thermodynamics: The Various Partition Functions I

47m 23s

Intro
0:00
Lesson Overview
0:19
Monatomic Ideal Gases
6:40
Monatomic Ideal Gases Overview
6:42
Finding the Parition Function of Translation
8:17
Finding the Parition Function of Electronics
13:29
Example: Na
17:42
Example: F
23:12
Energy Difference between the Ground State & the 1st Excited State
29:27
The Various Partition Functions for Monatomic Ideal Gases
32:20
Finding P
43:16
Going Back to U = (3/2) RT
46:20
Statistical Thermodynamics: The Various Partition Functions II

54m 9s

Intro
0:00
Diatomic Gases
0:16
Diatomic Gases
0:17
Zero-Energy Mark for Rotation
2:26
Zero-Energy Mark for Vibration
3:21
Zero-Energy Mark for Electronic
5:54
Vibration Partition Function
9:48
When Temperature is Very Low
14:00
When Temperature is Very High
15:22
Vibrational Component
18:48
Fraction of Molecules in the r Vibration State
21:00
Example: Fraction of Molecules in the r Vib. State
23:29
Rotation Partition Function
26:06
Heteronuclear & Homonuclear Diatomics
33:13
Energy & Heat Capacity
36:01
Fraction of Molecules in the J Rotational Level
39:20
Example: Fraction of Molecules in the J Rotational Level
40:32
Finding the Most Populated Level
44:07
Putting It All Together
46:06
Putting It All Together
46:07
Energy of Translation
51:51
Energy of Rotation
52:19
Energy of Vibration
52:42
Electronic Energy
53:35
XXV. Statistical Thermodynamics Example Problems
Example Problems I

48m 32s

Intro
0:00
Example I: Calculate the Fraction of Potassium Atoms in the First Excited Electronic State
0:10
Example II: Show That Each Translational Degree of Freedom Contributes R/2 to the Molar Heat Capacity
14:46
Example III: Calculate the Dissociation Energy
21:23
Example IV: Calculate the Vibrational Contribution to the Molar heat Capacity of Oxygen Gas at 500 K
25:46
Example V: Upper & Lower Quantum State
32:55
Example VI: Calculate the Relative Populations of the J=2 and J=1 Rotational States of the CO Molecule at 25°C
42:21
Example Problems II

57m 30s

Intro
0:00
Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels
0:10
Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature
8:05
Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity
11:59
Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF
45:03
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Lecture Comments (13)

3 answers

Last reply by: Professor Hovasapian
Sun Nov 18, 2018 4:18 AM

Post by peter alabi on December 29, 2017

Hi Dr. Hovasapian,
I just want to bring to your attention that the three previous lessons: Schr?dinger Equation & Operators, Schr?dinger Equation as an Eigenvalue Problem, and The Plausibility of the Schr?dinger Equation, are not playing. If there is something you can do about this, that I'll be great. Thanks.

1 answer

Last reply by: Professor Hovasapian
Wed Nov 25, 2015 12:53 AM

Post by Jupil Youn on November 18, 2015

When we are talking about probability density, we should indicate the interval such as dx.  Here is my question:  what is the probability of finding a particle at a given point?  I know dx is zero in this case. But dx is also infinitesimally small, isn't it?  Even if probability density function is continuous at a give interval, why we can not calculate prob at a specific point?

3 answers

Last reply by: Professor Hovasapian
Mon Mar 2, 2015 6:43 PM

Post by Frederic Hunt Hunt on February 7, 2015

Hello, I noticed that you mentioned viewing the appendix for a more in depth explanation. I was wondering where exactly is the appendix?

1 answer

Last reply by: Professor Hovasapian
Wed Nov 19, 2014 5:34 AM

Post by Joseph Szmulewicz on November 16, 2014

Sorry, I went over the part where you set the boundary equal to 0 and determine that A is equal to 0, which eliminates the A part when you set the boundary to a. So, I figured it out myself. thanks

0 answers

Post by Joseph Szmulewicz on November 16, 2014

You lost me on the part where you discuss the boundary equal to a. Why is the Acospa crossed out immediately, leaving the Bsinpa part only? You don't explain this

The Particle in a Box Part I

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Free Particle in a Box 0:28
    • Definition of a Free Particle in a Box
    • Amplitude of the Matter Wave
    • Intensity of the Wave
    • Probability Density
    • Probability that the Particle is Located Between x & dx
    • Probability that the Particle will be Found Between o & a
    • Wave Function & the Particle
    • Boundary Conditions
    • What Happened When There is No Constraint on the Particle
    • Diagrams
    • More on Probability Density
  • The Correspondence Principle 46:45
    • The Correspondence Principle
  • Normalizing the Wave Function 47:46
    • Normalizing the Wave Function
    • Normalized Wave Function & Normalization Constant

Transcription: The Particle in a Box Part I

Hello and welcome back to www.educator.com and welcome back to Physical Chemistry.0000

Today, we are going to talk about the particle in a box.0004

The particle in a box is a reasonably simple Quantum Mechanical problem.0007

It is our first dealing with a quantum mechanical system.0012

We are going to solve for the wave equation.0016

We are going to investigate the energy levels and things like that.0018

It is very important.0021

What we do here is going to set the pattern for what we continue to do throughout the quantum mechanics.0023

Let us get started.0027

Let us recall the Schrӧdinger equation.0031

I will stick with black that is not a problem.0033

We have –H ̅²/ 2 M D² DX² + this potential energy × the wave function.0038

Actually, I’m not going to put the X of this wave function.0053

I will leave it like that = the energy × the wave function.0056

This is the Schrӧdinger equation, this is the partial differential equation that needs to be,0062

in this case it is an ordinary differential equation because it is just a single variable X.0067

In general, it was a partial differential equations that needs to be solved for ψ.0070

What we are looking for when we solve this equation is this wave function.0075

What is it that wave function represents the particle?0081

Instead of dealing with it as a particle, we are thinking about the particle as a wave and0084

this wave function which is a function of X represents how the particle behaves in any given circumstance.0090

Now the box in this particle in the box, what we are talking about is the following.0099

We are going to be dealing first with the particle in a box of 1 dimension.0105

The box is exactly what you think it is, just think of a box and if I drop a particle in there, it is going to be a 3 dimensional box.0109

Or a 2 dimensional box is just a plane and it could be square, rectangle, whatever.0115

A 1 dimensional box is just an interval.0120

The box here in 1 dimension, we are going to start with a 1 dimensional problem and we will go ahead and extend it to 2 and 3.0123

In 1 dimension, it is just an interval, that is it.0133

It is just an interval on a pure line.0137

Let us say from 0 to A, that is our box.0143

The particle is going to be basically found somewhere in here.0149

It can only be there.0152

It cannot be out here, it cannot be out here that is all these means.0154

We are going to study the free particle constrained to lie between 0 and A.0160

The free particle in a box.0169

The word free particle here, free particle means it experiences no potential energy.0172

It experiences no potential energy.0189

In other words, V in the Schrӧdinger equation is going to be 0, in this particular case.0192

It simplifies our equation a little bit.0197

It experiences no potential energy.0199

Imagine just taking this particle, dropping it on this interval and saying where you are going to be.0204

How fast you are going to be moving.0211

Where is it go, things like that.0213

What it is going to do.0214

It can only do 1 of 2 things.0215

It basically can go this way or it can go this way.0217

The questions that we pose in quantum mechanics are which direction it is moving?0221

How fast is it moving at any given moment, can it tell you where it is?0225

Those are the questions that we want to ask.0229

These are the questions that the wave function is hopefully going to answer for us.0231

Free particle means it experiences no potential energy.0236

Which means that V of X = 0.0239

This equation actually ends up becoming the following.0244

It ends up becoming a - H ̅²/ 2 M.0247

The derivative of this squared, the second derivative of that = E × our wave function ψ.0252

I’m going to go ahead and rearrange this and write it in a way that is more convenient for solving the differential equation.0264

It is a way that you learn when you are taking the differential equation course.0274

And it will make sense in just a minute.0277

I’m going to rearrange this.0279

Basically, we multiply by the 2 M, I divide by that, bring everything over to 1 side, and set everything equal to 0.0280

It can look like this.0287

It is going to be D² ψ DX² + 2 M E/ H ̅² × ψ = 0.0289

And again, this wave function ψ, it is a function of X.0305

It is just a normal function like anything else, sin X, cos X, log of X.0310

That is all it is, that is what we are looking for.0320

In algebra, we have an equation like 2X + 3 = 5.0322

You are solving for X and you are trying to find a number.0326

A differential equation is the same thing except that it kicked up a couple of levels.0328

The variable that you are looking for is not a number, it is an actual function, that is all a differential equation is.0331

It is just a fancy algebraic equation.0337

In fact, there are techniques that actually reduce these straight to algebra.0341

Do not get lost in the fancy mathematics here.0348

It is just we are solving this, it is a little bit more complicated but we are just looking for some variable.0350

Our variable happens to be a function.0356

I’m going to leave off this X just to save some notation.0358

Again, we have to specify X between A and 0.0362

That is our constraints, they can only be between here and here.0367

We are putting the constraints on it.0374

The question is how can we interpret this ψ?0377

How can we interpret that?0382

We said that the wave function represents the amplitude of the matter wave in the previous lesson.0383

In classical mechanics, when we square the amplitude of the wave, we get the intensity of that wave.0415

The square of the amplitude which this is, represents the intensity of the wave.0423

I’m going to do it this way.0452

It is going to be ψ of X the complex conjugate × ψ.0454

We are not just going to do ψ X × ψ of X × ψ of X.0459

Just in case this wave function happens to be a complex function, we need to take the complex conjugate × ψ, in order to get a real quantity.0465

Anytime you have a given number that is complex, if you multiply the two conjugates together A + Bi A – Bi,0475

You are going to end up getting a real quantity.0482

We want a real quantity when we square this which is why this complex conjugate shows up.0484

If ψ happens to actually end up being a real function like cos of X that it is not a problem.0489

The square is just cos² of X because the conjugate of something real is the thing itself.0495

The conjugate of 5 is 5, it is not a problem when we place that conjugate there, simply just in case ψ is complex.0501

The square of the amplitude represents the intensity of the wave.0513

We have this thing.0516

Our problem is how the heck we are going to interpret intensity?0518

What intensity mean when it comes to a particle?0520

How do we interpret intensity?0525

Here is how we do it.0527

Think of intensity as the extent to which a particle is actually present.0530

If a particle has a wave function and the square of that wave function which is the intensity is kind of low,0536

That means that chances are really low that you actually find a particle there.0545

If the intensity is really high, then, in other words if the square of the wave function is high then0552

that means chances are really good that you are going to find a particle there.0560

We are going to interpret intensity as a probability that a particular particle is at a given place.0565

We are going to express it as a probability, lower the intensity, lower the probability.0573

Higher the intensity, higher the probability.0578

We have ψ conjugate × ψ is the probability density.0581

We do not need to concern ourselves too much with the probability density so much0596

because we are going to be integrating this thing and we would be concerned with probabilities.0602

That is what is going to be most important.0606

With the probability density, it is like any other density.0608

If there is a mass density that is a certain mass per volume, g/ ml.0611

The probability density is a certain probability per volume, or per length element.0618

In this particular case, we are sticking to 1 dimension.0626

If I take a little differential element DX like that, the product of the conjugate a ψ × ψ itself.0629

And that giving me the probability density.0642

It gives me the probability per unit of length or unit of area or unit of volume.0646

That is all it is, it is like any other density.0652

Our quantity is this, it is ψ and ψ × DX.0657

This right here, when I take the wave function conjugate × wave function,0666

I will just say the square of the wave function.0671

I will just say it that way.0673

The square of the function × some differential length element, this is the actual probability.0674

This is the probability that the particle is located between a given X and DX.0683

Let me redraw this thing right here.0705

If I have some value of X, if I take some differential length DX,0710

Now this is X + DX.0717

If I take the square of the wave function and multiply it by this length element,0720

I end up actually getting the probability that the particle is located here in that differential element.0724

If I integrate, all of the differential elements from 0 to A, I get the probability that the particle is somewhere between 0 and A.0729

That is the whole idea.0741

What is important is the square × the DX, that is the probability.0743

When I just take the square of the wave function, I get something called the probability density.0746

It is good to know but is not going to get in our way too much.0751

Let us go ahead and do the integration.0756

When I integrate from 0 to A, the square of the wave function.0759

This is basically I'm just adding all the probabilities.0769

The probability of the particles here or here.0775

All the DX is just like normal integration and mathematics.0780

When I add them all up, because probabilities are additive, I get the total probability that a particle is between 0 and A.0784

The probability that the particle will be found between 0 and A.0795

Since, I'm saying that the particle is going to be somewhere between 0 and A,0816

I can say for sure that somewhere between 0 and A, I’m going to find the particle.0821

I may not know where exactly it is, but I know it is going to be between 0 and A.0825

My probability is 100% it is equal to 1.0829

This thing is actually going to end up equaling 1.0832

We will see a little bit more of that in just a little bit.0836

Here, what is important is that the square of the wave function = the intensity.0838

We are going to interpret the intensity as the probability of finding the particle there.0843

The square of the wave function × some differential element, whether it is a length and area, or a volume,0850

it gives me the probability that the particle is located between or in that differential length element or the area element or volume element.0855

When I integrate that over my particular interval, I get the probability that the particle would be found between 0 and A.0864

Whenever I specify A to B, whatever my interval is.0872

This is what is important right here, profoundly important.0874

That is how we are going to interpret this wave function.0880

It is the square of it represents a probability density and the square × the differential element is the actual probability.0885

Let us concern ourselves with the ψ.0897

This wave function represents a particle, we know that.0900

What we have is this thing right here.0912

Between 0 and A, ψ of X represents a particle.0921

We are constraining you to lie between 0 and A so we know for sure that is not going to be to the left of 0, it is not going to be to the right of A.0927

Over here, our wave function is going to equal 0.0934

And over here, our wave function is going to equal 0.0939

Since we are differentiating not once but twice, the Schrӧdinger equation is a second derivative.0946

In order to take the derivative of this function, the function needs to be continuous.0953

Because it is 0 outside of the interval over here, 0 over here, and 0 over here, because the function is continuous,0958

Because we have to differentiate it, that means it has to be 0 actually at 0 and at A.0968

In other words, you are not going to have some wave function.0975

Let us say this is 0 and this is A, you are not going to have some wave function that goes like this.0978

It is all 0 from here, there is a discontinuity here.0983

That is not going to happen.0987

It needs to be 0 here because it is 0 pass those points.0988

Therefore, at 0 and A, the wave function ψ of 0 has to equal 0.0995

The ψ of A has to equal 0.1009

These things are called boundary conditions.1013

This is another set of constraints that we have to place on the particular problem in order for it to actually make sense.1024

The wave function has to be 0 at 0, it is 0 at A.1029

You are specifying what is happening at the boundary.1036

We just want to find out what is going on in between.1038

Our mathematical problem becomes something called a boundary value problem.1042

Here is what we are going to do.1046

We have to solve the equation + 2 ME/ H ̅² × ψ = 0,1048

Where X ≥ 0, ≤ A subject to the boundary conditions ψ of 0 = 0 and ψ of A = 0.1063

This is the mathematical problem that we have to solve.1076

We have to solve this thing subject to these constraints.1079

Particularly this, the boundary conditions.1083

Let us go ahead and do it.1087

As far as the solution of this going through this, you can take a look at the appendix1090

if you actually want to see how one goes through solving this particular differential equation.1094

Here I’m just going to go ahead and present the solutions.1098

And the solutions are really all that we need.1100

Only if you want the extra information, you are welcome to look at it.1102

Here, when we solve this we get the following.1107

We get the general solution is A × cos of PX + B × sin of PX,1111

Where P, I just put a P in here to make it a little easier instead of writing out everything.1126

Where P is actually equal to 2 ME¹/2, make it a radical sign if you want, / H ̅.1132

If you want, you can put this in here and here.1141

I just decided to call it P or you can call it whatever it is that you want.1144

We call the H ̅ is equal to H/ 2 π.1149

This is our general solution.1156

We found that the ψ, we found a way function.1158

Let us subject that wave function to our boundary conditions and see what A and B are going to be.1162

Now the boundary conditions.1169

This is normally how you handle all differential equations.1171

You solve the equation and then you take a look at whatever constraints that you have placed on it1173

to find the values of the individual constants.1178

Instead of the general solution, you try to find a specific solution or a specific set of solutions.1181

Now the boundary conditions.1190

Let us go ahead and deal with the first one.1199

I'm going to go to blue here.1200

Let us go ahead and deal with this one.1203

The ψ of 0 = A × I just put it into the equation, I see what I get,1206

A × cos of P × 0 because we are putting in for X, + B × sin of P × 0 = 0.1216

The sin 0 is 0, the cos of 0 is 1.1230

What we get here is A × 1 and this is 0, A × 1 = 0.1236

Therefore, A = 0.1243

We found what A is, it is equal to 0.1245

We will go ahead and this term just drops out.1248

We will go ahead and deal with the second boundary condition, that the ψ of A = 0.1252

Since we know that we are not dealing with this term anymore, we have B × the sin of P.1260

A, X is we are putting A in and that is also equal to 0.1269

There are two things that can happen, N can be 0 or sin of PA can be 0.1277

B equal to 0 is trivial so it does not give us anything.1281

We do not have to worry about that.1284

However, let us deal with the sin of PA equaling 0.1286

Which means that PA = the inverse sin of 0.1294

And what you end up getting here is PA = 0, you get π, you get 2π, you get 3π, 4π, and so on.1300

I’m going to do PA, and I’m going to write it as Nπ.1311

Where N is going to equal 1, 2, 3 and so on.1319

I’m not going to conclude the 0 because again it does not really give us anything.1324

Here is what we end up getting.1328

PA = Nπ.1330

N= 1, 2, 3, all of these make this boundary condition true.1333

Let us go to the next page here.1353

We said P was equal to 2 ME¹/2/ H ̅.1356

And we just said that PA = Nπ.1373

Let us put this in for P so what we end up getting is 2 ME¹/2/ H ̅.1377

I hope I’m not confusing my H and H ̅, that is a big problem here in Quantum Mechanics.1388

It is equal to = N π.1394

I’m going to rearrange this equation, when you rearrange and solve for E, the energy, you end up with the following.1396

You end up with E for a given N because N is 1234.1403

It is going to equal H ̅² N² I²/ A² 2M.1410

Since, H ̅ = H/ 2 π which implies that H ̅² = H²/ 4 π²,1432

An alternative version involving planks constant directly instead of H ̅ is going to be H² N²/ A² 8 M.1447

And remember, M is the mass of the particle.1463

What you have is, this is the energy of the particle, either you are this one or this one.1465

It is totally up to you.1474

Notice that the energy of the particle is quantize.1476

It has very specific values depending on what N is.1479

When N is 1, it has a certain energy.1483

When N is 2, it has a certain energy.1485

When N is 3, it has a certain energy.1488

N is integral 12345, there is no 1.5, 1.6, 1.7, radical 14, things like that.1491

This is what we mean by quantization.1500

In other words, this particle in the box, the energy of the particle could only have specific values.1503

There is no in between.1511

It cannot take any value that it wants to. This is very different than classical behavior.1513

Classical particle that have any energy value at all depending on what is going on.1518

It does not matter.1523

Here it is very specific and this energy is actually contingent on N, some quantum numbers, some integer.1525

This N here is called a quantum number.1534

This is a constant, that is a constant, A is whatever you happen to chose.1536

You can change it but once you choose it, it is fixed amount.1540

That is the length to M.1544

All of these are constants, this energy is a function of N some number that1547

shows up out of nowhere simply by virtue of the solution to the differential equation.1551

This is what is extraordinary about quantum mechanics.1555

This is what we mean by the quantum.1558

The energy is quantized.1560

N is called a quantum number.1563

Notice, this quantum number shows up naturally by virtue of our solution to the problem.1574

This is going to be a running theme in quantum mechanics.1580

There is going to be several quantum numbers that show up based on how we solve problem.1583

Let us see what we have got.1590

Well, PA = N π.1591

Therefore, P is actually equal to N π/ A.1598

Therefore, we can go ahead and write our ψ sub N of X depending on what N is.1605

It is equal to B × sin N π/ A × X and the energy for a given N = DH² N²/ A² 8M.1614

There we go.1635

You are probably saying to yourself what is B?1638

We have not found out what B is, you are right.1641

We will find out what B is in just a moment.1642

But this is the solution to our particular particle in a box.1646

For different numbers 1234567, the wave function is this thing, it represents the particle.1651

Any information that I want about the particle, I'm going to extract from that function.1659

If I want to know the energy for any given state 12345, I just plug it into here and I get the energy of the particle.1664

That is what is happening.1672

What would happen if we did not impose any constraints on the particle?1678

If we just said, here is a free particle, it is not experiencing any potential energy, tell me something about its energy?1684

What is happening?1691

If we did not place any restraints on the particle, in other words if we did not restrict the particle to lie between 0 and A,1694

Instead, the particle can be anywhere.1701

Mathematically it means that the ψ of 0, if we do not strain it to lie between 0 and A, the most boundary conditions that ψ of 0 = 0.1703

The ψ of A is equal 0, they vanish.1726

All you are left with is the solution to the mathematical equation.1728

The problem ends up just being the differential equation without the boundary value problems.1732

I end up without the boundary conditions.1738

Once we solve that, you end up getting the same solution.1741

It is the same differential equation.1745

You end up with something like ψ of X = A × cos of P of X + B × sin of P of X.1747

P is the same thing where P = √2 ME/ H ̅.1760

Now since there are no boundary conditions, I'm not constrained.1771

I do not end up having to find this A and this B.1779

I do not end up having to find this PA = something.1782

It turns out that P can actually be anything.1785

Because P can be anything, because P can be any number at all, before, we had P = N π/ A.1790

P depended on N.1804

There are no boundary conditions for a free particle completely.1806

That is not a particle in a box, it is free to move anywhere it want.1811

It is not dependent on N, P just equals this.1815

When I rearrange this for energy, the energy can actually take on any value at all.1819

And that is what is interesting here.1824

What you end up getting when you rearrange this, you end up with E = P² H ̅²/ 2 M.1825

I hope that the mathematics is properly here.1836

But the idea is that P is no longer contingent on N.1840

It can be anything.1844

Therefore, the energy can be anything.1846

Now the energy of a completely free particle that is not constrained to lie in a particular region, it can take on any energy value at all.1847

In other words, it is not quantized.1857

Quantization, this whole idea of quantum mechanics, the quantum property of a particle,1860

only appears by virtue of the constraints that we place upon our particular system.1866

When we remove those constraints, it allow a particle just do what it wants to do, whenever it wants to do and however it wants to do it.1872

Everything is fine, everything behaves normally.1878

The energy is not quantized at all.1881

It can take on any value just like a normal classical particle.1883

That is pretty extraordinary.1886

Quantization appears only when we begin to place constraints on a given system.1888

What this means, in the case of the particle in a box, the constraints that we put on it, this particle wave,1894

what we are saying is that it has to fit inside the box.1919

It is what quantization means.1930

That means there are only certain waves that will satisfy this fit property that I draw in just a second.1932

If I do not place any constraints then it can be any wave that all.1940

It doe not really matter, that is right there.1942

It can be any wave at all.1944

When I place constraints on it, it can only be specific waves, waves that fit into that box.1946

And because it can be only specific waves, those waves can have only specific energy values.1951

That is what is happening.1956

Quantization is an emergent property.1958

It is something that comes about by placing constraints on the system.1960

Let us fit into a box, inside the box, such that as we said our ψ of 0 = 0 and our ψ of A equal 0.1966

Drawing wise, it means this.1980

If this is 0 and A, that is one possibility for the wave.1983

0 and A, that is another possibility for the wave.1990

Notice, I have to be the wave that has to begin and end there.1996

This is 0 and this is A, begins and ends there.2003

What you are not going to see is something like this.2013

This is 0 and this is A.2019

It is not going to be up here.2021

It is not going to be down here.2023

It is not just some random wave, very specific waves with very specific energies, the constraint.2025

Quantum behavior emerges as a result to the constraints that we place it on the system.2033

The more constraints we place upon the system, the more restricted we are in the particular values that the energy of the particle can be.2039

Let us go ahead and draw this formally here.2051

Let us go ahead and this is going to be N and I'm going to start at 1, 2, 3, and 4.2055

I have this one, this is 0 to A, 0 to A.2068

This is 0 and this is A.2080

This is going to be our ψ sub 1, our ψ sub 2, our ψ sub 3, our ψ sub 4, and so on.2084

I will just do the first four.2092

Over here, I’m going to go ahead and here I’m going to draw the wave function.2094

Here I’m going to draw the probability density, the square of the wave function.2100

It is that.2107

This is going to be ψ sub 1 conjugate, ψ sub 1, ψ sub 2 conjugate, ψ sub 2, ψ sub 3 conjugate, ψ sub 3 and ψ sub 4 conjugate, ψ sub 4.2118

That is our first wave when we have the B sin N π/ X.2139

Our ψ sub 1.2148

Let us try this again.2150

Our ψ sub N of X we said was equal to B sin N π/ A × X.2160

If N = 1, we end up with B sin π/ A × X.2170

We have B sin 2 π/ A × X, it is going to be this one.2174

Here and here, we have that.2193

1 node, 2 node, 3 node, and so on.2203

We have this and this.2208

Begins and ends at A.2219

This is the wave function.2221

Notice, they are just normal standing waves.2223

You have a string that you are holding at one end, a string that you are holding at the other end,2226

You pluck that string, it is going to vibrate in different frequencies.2231

1 frequency, 1 frequency, another frequency and another frequency, they represent the different and the N you have different energy.2237

There is certain energy.2244

That certain energy you get from the equation that we saw.2246

When I square these wave functions, I get the probability density or I can just think of it as the probability at this point.2252

When I square this, this one looks like this.2260

What this means is that the particles more likely to be found here towards the center than it is to be found here.2266

Notice, the probability, the intensity of the wave is lower here.2273

There is less of a chance that I’m going to find a particle here or here.2276

When a particular particle was in this state, it is in the 1 state, the probability I’m going to find it here.2281

This one ends up being, when I square it, it is going to look like this.2286

The probability of finding the particle at the center is 0.2297

More than likely that I’m going to find the particle here or here.2299

That is what is going on here.2303

And so on, and so forth.2313

These high points are the greatest probability of finding the particles.2322

If there is our particle happens to be in the state 3, the probability of finding the particle here or here or here.2329

There is very little probability that I will find it here or here, and so forth.2338

Notice, here mostly it is concentrated in the center.2342

Here it is off to the sides.2345

Here it is a little bit more distributed.2346

Here it is a little bit more distributed evenly.2348

As N gets bigger and bigger, the distribution of the particle actually becomes a little bit more uniform.2351

That is what is happening here.2359

This is just a pictorial representation of the wave function and the probability density.2361

In other words, where you are going to find the particle.2367

The high points have the highest probability of finding the particle.2370

As you get lower and lower, there is a lower probability of finding the particle there.2373

That is all that is going on here.2380

Since this thing is actually a real function, in this particular case, we do not need the conjugate2385

but we will go ahead and use it because that is the symbolism.2398

This × that = ψ is this.2400

Ψ conjugate is also that because this is a real function, it is not a complex function.2409

The product actually = B² sin² N π/ A × X.2416

That is it, we are just multiplying this by itself because it is a real quantity so the conjugate is just ψ.2423

Since, ψ of X is real, my conjugate happens to equal ψ.2435

When I do that, I get that.2445

Let us say a little bit more here.2452

The probability density, the second graph.2455

The probability density for N = 1, it shows that a particle is most likely to be found near the center.2465

It is most likely to be found near the center of the interval.2484

For N = 2 it is more likely to be found near A/ 4 or 3A/ 4.2494

Those are high points.2522

As N increases, the particles are more likely to be found more uniformly distributed across the interval.2525

In other words, here is what happens.2563

As N gets bigger and bigger, let us say for something like N = 30.2566

For something like N= 30, you do have something looks like this.2575

We have this and this is 0, this is A, you are going to have something like.2584

The distribution becomes broader.2595

In other words, now you are very likely to find it almost anywhere where there is a high point.2597

Now you are not restricted.2603

As N increases, that particle is more likely to be found more uniformly distributed across the interval.2606

This is how a normal classical particle would behave.2616

In other words, when we treat a particle the way we have treated it in the physics courses2618

that you have taken as a particle, another wave, the particle can be anywhere.2622

It has no preference.2627

It can be absolutely anywhere.2631

When we treat it like a wave, now it has a preference.2633

It has a greater probability of being here or here or here.2637

As N increases, these high points tend to also increase.2642

Now, you end up finding it more and more places.2647

As N increases, it starts to behave like a classical particle.2651

That is what is going on.2658

As N increases, the probability density or the probability becomes more uniform which is how a classical particle behaves.2661

In other words, it shows no preference for where it is in the interval.2704

It is just as likely to be here, as it is here.2717

When we are looking at the N = 1, it is something that looks like this.2732

Not like that, it is a little bit worse.2741

We have something that looks like that.2746

Here the particle shows a preference.2750

The probability density, the square of the wave function when we graph it,2753

it shows that the particles more likely to be here somewhere in this area, that is not so likely to be here.2756

That is really interesting.2763

Under quantum behavior, it actually has a preference for where it is going to be.2766

As N increases, you get more and more.2772

It can be here, here, or here.2775

Now, it spends more time everywhere, that is the idea.2777

It is spending more time everywhere.2782

As N increases, the distribution becomes more uniform.2784

Here, this is not a uniform distribution.2788

Basically, it is going to be like somewhere in this region.2790

It is going to avoid this area and this area.2793

As N increases, it starts to show the behavior of a classical particle.2796

This illustrates something called the correspondence principle.2802

We will see this again, this correspondence principle.2806

Quantum mechanical results, they approach classical mechanics as quantum numbers get bigger.2818

As the quantum numbers get bigger, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40.2852

The particle starts to behave more like a classical particle.2861

Let us talk about something very important called normalizing the wave function.2864

Let me go back to black for this one and now we are going to go ahead and figure out what that B is.2873

Normalizing the wave function.2881

We said that our ψ sub N of X = B × sin N π/ A × X.2893

This is our wave function for a particle in a box.2902

We are going to take the ψ conjugate × ψ × DX = B² sin² N π AX × DX.2908

This is the probability that the particle will be found in the interval DX, wherever happen to take that DX.2924

We are restricting the particle to lie between 0 and A, the probability that2933

a particle is going to be found somewhere between 0 and A is 100% it is equal to 1.2943

Therefore, when I integrate this probability density over the entire interval from 0 all the way to A, I get the total probability.2949

The probability that I'm going to find the particle somewhere between 0 and A.2971

Again, the square of the wave function × DX is the probability that I'm going to find the particle in that little interval,2976

that little differential interval DX.2983

When I integrate over the entire interval from 0 to A, I get the probability that the particles can be found between 0 and A.2986

I know I’m going to find it somewhere.2993

Therefore, this is equal to 1.2996

Let us go ahead and solve this.3002

I’m going to pull the B² out.3004

When I go ahead and I solve this integral, you can do it either with mathematical software that you have,3019

your mathematical or maple, or mathcad, whatever is that you happen to be using.3024

You can solve this by looking in a table of integrals online.3029

You have table of integrals in the back of your first year calculus text.3033

You have table of integrals in the CRZ handbook.3037

I'm not going to bother with the actual integration all that much.3039

I’m going to be concerned with setting up the integral.3043

Mostly, I will just use software or tables to do the integrals.3046

I'm not going to go through the process.3050

When we solve this integral right here, we actually end up getting this.3052

It is going to be B² × A/ 2 = 1.3065

Therefore, we end up with.3079

I will stay on the same page.3081

I get B² = 2/ A.3083

Therefore, B = 2/ A ^½ or √2/ A.3090

We went ahead and we found B by using this property of probabilities that we know.3100

Now, we can write the individual wave function ψ sub N, they are equal to 2/ A¹/2 × sin of N π/ A × X.3107

X ≤ A, ≥ 0.3127

Energy sub N = A² N² / A² 8M.3130

This is the final solution to our particle in a 1 dimensional box.3141

A wave function that satisfies the integral when I have a particular wave function,3147

If I multiply it by a complex conjugate and I integrated it over the particular interval,3155

If I end up getting 1, that way function is said to be normalized.3173

A wave function that satisfies this relation is said to be normalized.3179

What we did was use this normalization condition to actually find B which was what we call this B²/ A ^½.3189

In this particular case, it is called a normalization constant.3199

If a function is the solution to a differential equation, any constant × that function is also a solution to the differential equations,3201

Because the Hamiltonian operator is linear, we are dealing with linear operators.3209

If F of X is a solution to a function then K × F of X.3215

Any constant × F of X is also a solution to that differential equation.3220

That is what is nice about this.3223

We can always adjust the constant which is what we did to make this happen.3225

If I take this function and if I multiply it by its conjugate which is just multiplying it by itself,3232

And if I integrate it from 0 to A, I’m going to get 1.3239

I have normalized this wave function.3243

We used this condition to find B which is called the normalization constant.3249

This is very important, normalization constant and a normalized wave function.3280

I will close it out with the following.3292

Because this DX is the probability of finding the particle between X and DX,3294

the integral from any X sub 1 to any X sub 2 within a particular interval of ψ conjugate × ψ × DX,3322

It gives the probability of finding the particle within that interval.3334

Again, this thing is the probability of finding it within that particular DX.3339

You can integrate from anywhere.3344

You do not have to integrate from 0 to A, that gives you the probability of finding it over the whole interval.3346

You can take a piece of the interval, you just change your upper and lower limits of integration.3351

This gives the probability of finding the particle between X1 and X2.3355

Thank you so much for joining us here at www.educator.com.3379

We will see you next time, bye.3381

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