Raffi Hovasapian

Schrӧdinger Equation as an Eigenvalue Problem

Slide Duration:Table of Contents

46m 5s

- Intro0:00
- Course Overview0:16
- Thermodynamics & Classical Thermodynamics0:17
- Structure of the Course1:30
- The Ideal Gas Law3:06
- Ideal Gas Law: PV=nRT3:07
- Units of Pressure4:51
- Manipulating Units5:52
- Atmosphere : atm8:15
- Millimeter of Mercury: mm Hg8:48
- SI Unit of Volume9:32
- SI Unit of Temperature10:32
- Value of R (Gas Constant): Pv = nRT10:51
- Extensive and Intensive Variables (Properties)15:23
- Intensive Property15:52
- Extensive Property16:30
- Example: Extensive and Intensive Variables18:20
- Ideal Gas Law19:24
- Ideal Gas Law with Intensive Variables19:25
- Graphing Equations23:51
- Hold T Constant & Graph P vs. V23:52
- Hold P Constant & Graph V vs. T31:08
- Hold V Constant & Graph P vs. T34:38
- Isochores or Isometrics37:08
- More on the V vs. T Graph39:46
- More on the P vs. V Graph42:06
- Ideal Gas Law at Low Pressure & High Temperature44:26
- Ideal Gas Law at High Pressure & Low Temperature45:16

46m 2s

- Intro0:00
- Math Lesson 1: Partial Differentiation0:38
- Overview0:39
- Example I3:00
- Example II6:33
- Example III9:52
- Example IV17:26
- Differential & Derivative21:44
- What Does It Mean?21:45
- Total Differential (or Total Derivative)30:16
- Net Change in Pressure (P)33:58
- General Equation for Total Differential38:12
- Example 5: Total Differential39:28

1h 6m 45s

- Intro0:00
- Properties of Thermodynamic State1:38
- Big Picture: 3 Properties of Thermodynamic State1:39
- Enthalpy & Free Energy3:30
- Associated Law4:40
- Energy & the First Law of Thermodynamics7:13
- System & Its Surrounding Separated by a Boundary7:14
- In Other Cases the Boundary is Less Clear10:47
- State of a System12:37
- State of a System12:38
- Change in State14:00
- Path for a Change in State14:57
- Example: State of a System15:46
- Open, Close, and Isolated System18:26
- Open System18:27
- Closed System19:02
- Isolated System19:22
- Important Questions20:38
- Important Questions20:39
- Work & Heat22:50
- Definition of Work23:33
- Properties of Work25:34
- Definition of Heat32:16
- Properties of Heat34:49
- Experiment #142:23
- Experiment #247:00
- More on Work & Heat54:50
- More on Work & Heat54:51
- Conventions for Heat & Work1:00:50
- Convention for Heat1:02:40
- Convention for Work1:04:24
- Schematic Representation1:05:00

1h 6m 33s

- Intro0:00
- The First Law of Thermodynamics0:53
- The First Law of Thermodynamics0:54
- 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 Ways11:56
- Systems Possess Energy, Not Heat or Work12:45
- Scenario 116:00
- Scenario 216:46
- State Property, Path Properties, and Path Functions18:10
- Pressure-Volume Work22:36
- When a System Changes22:37
- Gas Expands24:06
- Gas is Compressed25:13
- Pressure Volume Diagram: Analyzing Expansion27:17
- What if We do the Same Expansion in Two Stages?35:22
- Multistage Expansion43:58
- General Expression for the Pressure-Volume Work46:59
- Upper Limit of Isothermal Expansion50:00
- Expression for the Work Done in an Isothermal Expansion52:45
- Example 2: Find an Expression for the Maximum Work Done by an Ideal Gas upon Isothermal Expansion56:18
- Example 3: Calculate the External Pressure and Work Done58:50

1h 2m 17s

- Intro0:00
- Compression0:20
- Compression Overview0:34
- Single-stage compression vs. 2-stage Compression2:16
- Multi-stage Compression8:40
- Example I: Compression14:47
- Example 1: Single-stage Compression14:47
- Example 1: 2-stage Compression20:07
- Example 1: Absolute Minimum26:37
- More on Compression32:55
- Isothermal Expansion & Compression32:56
- External & Internal Pressure of the System35:18
- Reversible & Irreversible Processes37:32
- Process 1: Overview38:57
- Process 2: Overview39:36
- Process 1: Analysis40:42
- Process 2: Analysis45:29
- Reversible Process50:03
- Isothermal Expansion and Compression54:31
- Example II: Reversible Isothermal Compression of a Van der Waals Gas58:10
- Example 2: Reversible Isothermal Compression of a Van der Waals Gas58:11

1h 4m 39s

- Intro0:00
- Recall0:37
- State Function & Path Function0:38
- First Law2:11
- Exact & Inexact Differential2:12
- Where Does (∆U = Q - W) or dU = dQ - dU Come from?8:54
- Cyclic Integrals of Path and State Functions8:55
- Our Empirical Experience of the First Law12:31
- ∆U = Q - W18:42
- Relations between Changes in Properties and Energy22:24
- Relations between Changes in Properties and Energy22:25
- Rate of Change of Energy per Unit Change in Temperature29:54
- Rate of Change of Energy per Unit Change in Volume at Constant Temperature32:39
- Total Differential Equation34:38
- Constant Volume41:08
- If Volume Remains Constant, then dV = 041:09
- Constant Volume Heat Capacity45:22
- Constant Volume Integrated48:14
- Increase & Decrease in Energy of the System54:19
- Example 1: ∆U and Qv57:43
- Important Equations1:02:06

16m 50s

- Intro0:00
- Joule's Experiment0:09
- Joule's Experiment1:20
- Interpretation of the Result4:42
- The Gas Expands Against No External Pressure4:43
- Temperature of the Surrounding Does Not Change6:20
- System & Surrounding7:04
- Joule's Law10:44
- More on Joule's Experiment11:08
- Later Experiment12:38
- Dealing with the 2nd Law & Its Mathematical Consequences13:52

43m 40s

- Intro0:00
- Changes in Energy & State: Constant Pressure0:20
- Integrating with Constant Pressure0:35
- Defining the New State Function6:24
- Heat & Enthalpy of the System at Constant Pressure8:54
- Finding ∆U12:10
- dH15:28
- Constant Pressure Heat Capacity18:08
- Important Equations25:44
- Important Equations25:45
- Important Equations at Constant Pressure27:32
- Example I: Change in Enthalpy (∆H)28:53
- Example II: Change in Internal Energy (∆U)34:19

32m 23s

- Intro0:00
- The Relationship Between Cp & Cv0:21
- For a Constant Volume Process No Work is Done0:22
- For a Constant Pressure Process ∆V ≠ 0, so Work is Done1:16
- The Relationship Between Cp & Cv: For an Ideal Gas3:26
- The Relationship Between Cp & Cv: In Terms of Molar heat Capacities5:44
- Heat Capacity Can Have an Infinite # of Values7:14
- The Relationship Between Cp & Cv11:20
- When Cp is Greater than Cv17:13
- 2nd Term18:10
- 1st Term19:20
- Constant P Process: 3 Parts22:36
- Part 123:45
- Part 224:10
- Part 324:46
- Define : γ = (Cp/Cv)28:06
- For Gases28:36
- For Liquids29:04
- For an Ideal Gas30:46

39m 15s

- Intro0:00
- General Equations0:13
- Recall0:14
- How Does Enthalpy of a System Change Upon a Unit Change in Pressure?2:58
- For Liquids & Solids12:11
- For Ideal Gases14:08
- For Real Gases16:58
- The Joule Thompson Experiment18:37
- The Joule Thompson Experiment Setup18:38
- The Flow in 2 Stages22:54
- Work Equation for the Joule Thompson Experiment24:14
- Insulated Pipe26:33
- Joule-Thompson Coefficient29:50
- Changing Temperature & Pressure in Such a Way that Enthalpy Remains Constant31:44
- Joule Thompson Inversion Temperature36:26
- Positive & Negative Joule-Thompson Coefficient36:27
- Joule Thompson Inversion Temperature37:22
- Inversion Temperature of Hydrogen Gas37:59

35m 52s

- Intro0:00
- Adiabatic Changes of State0:10
- Adiabatic Changes of State0:18
- Work & Energy in an Adiabatic Process3:44
- Pressure-Volume Work7:43
- Adiabatic Changes for an Ideal Gas9:23
- Adiabatic Changes for an Ideal Gas9:24
- Equation for a Fixed Change in Volume11:20
- Maximum & Minimum Values of Temperature14:20
- Adiabatic Path18:08
- Adiabatic Path Diagram18:09
- Reversible Adiabatic Expansion21:54
- Reversible Adiabatic Compression22:34
- Fundamental Relationship Equation for an Ideal Gas Under Adiabatic Expansion25:00
- More on the Equation28:20
- Important Equations32:16
- Important Adiabatic Equation32:17
- Reversible Adiabatic Change of State Equation33:02

42m 40s

- Intro0:00
- Fundamental Equations0:56
- Work2:40
- Energy (1st Law)3:10
- Definition of Enthalpy3:44
- Heat capacity Definitions4:06
- The Mathematics6:35
- Fundamental Concepts8:13
- Isothermal8:20
- Adiabatic8:54
- Isobaric9:25
- Isometric9:48
- Ideal Gases10:14
- Example I12:08
- Example I: Conventions12:44
- Example I: Part A15:30
- Example I: Part B18:24
- Example I: Part C19:53
- Example II: What is the Heat Capacity of the System?21:49
- Example III: Find Q, W, ∆U & ∆H for this Change of State24:15
- Example IV: Find Q, W, ∆U & ∆H31:37
- Example V: Find Q, W, ∆U & ∆H38:20

1h 23s

- Intro0:00
- Example I0:11
- Example I: Finding ∆U1:49
- Example I: Finding W6:22
- Example I: Finding Q11:23
- Example I: Finding ∆H16:09
- Example I: Summary17:07
- Example II21:16
- Example II: Finding W22:42
- Example II: Finding ∆H27:48
- Example II: Finding Q30:58
- Example II: Finding ∆U31:30
- Example III33:33
- Example III: Finding ∆U, Q & W33:34
- Example III: Finding ∆H38:07
- Example IV41:50
- Example IV: Finding ∆U41:51
- Example IV: Finding ∆H45:42
- Example V49:31
- Example V: Finding W49:32
- Example V: Finding ∆U55:26
- Example V: Finding Q56:26
- Example V: Finding ∆H56:55

44m 34s

- Intro0:00
- Example I0:15
- Example I: Finding the Final Temperature3:40
- Example I: Finding Q8:04
- Example I: Finding ∆U8:25
- Example I: Finding W9:08
- Example I: Finding ∆H9:51
- Example II11:27
- Example II: Finding the Final Temperature11:28
- Example II: Finding ∆U21:25
- Example II: Finding W & Q22:14
- Example II: Finding ∆H23:03
- Example III24:38
- Example III: Finding the Final Temperature24:39
- Example III: Finding W, ∆U, and Q27:43
- Example III: Finding ∆H28:04
- Example IV29:23
- Example IV: Finding ∆U, W, and Q25:36
- Example IV: Finding ∆H31:33
- Example V32:24
- Example V: Finding the Final Temperature33:32
- Example V: Finding ∆U39:31
- Example V: Finding W40:17
- Example V: First Way of Finding ∆H41:10
- Example V: Second Way of Finding ∆H42:10

59m 7s

- Intro0:00
- Example I: Find ∆H° for the Following Reaction0:42
- Example II: Calculate the ∆U° for the Reaction in Example I5:33
- Example III: Calculate the Heat of Formation of NH₃ at 298 K14:23
- Example IV32:15
- Part A: Calculate the Heat of Vaporization of Water at 25°C33:49
- Part B: Calculate the Work Done in Vaporizing 2 Mols of Water at 25°C Under a Constant Pressure of 1 atm35:26
- Part C: Find ∆U for the Vaporization of Water at 25°C41:00
- Part D: Find the Enthalpy of Vaporization of Water at 100°C43:12
- Example V49:24
- Part A: Constant Temperature & Increasing Pressure50:25
- Part B: Increasing temperature & Constant Pressure56:20

49m 16s

- Intro0:00
- Entropy, Part 10:16
- Coefficient of Thermal Expansion (Isobaric)0:38
- Coefficient of Compressibility (Isothermal)1:25
- Relative Increase & Relative Decrease2:16
- More on α4:40
- More on κ8:38
- Entropy, Part 211:04
- Definition of Entropy12:54
- Differential Change in Entropy & the Reversible Path20:08
- State Property of the System28:26
- Entropy Changes Under Isothermal Conditions35:00
- Recall: Heating Curve41:05
- Some Phase Changes Take Place Under Constant Pressure44:07
- Example I: Finding ∆S for a Phase Change46:05

33m 59s

- Intro0:00
- Math Lesson II0:46
- Let F(x,y) = x²y³0:47
- Total Differential3:34
- Total Differential Expression6:06
- Example 19:24
- More on Math Expression13:26
- Exact Total Differential Expression13:27
- Exact Differentials19:50
- Inexact Differentials20:20
- The Cyclic Rule21:06
- The Cyclic Rule21:07
- Example 227:58

54m 37s

- Intro0:00
- Entropy As a Function of Temperature & Volume0:14
- Fundamental Equation of Thermodynamics1:16
- Things to Notice9:10
- Entropy As a Function of Temperature & Volume14:47
- Temperature-dependence of Entropy24:00
- Example I26:19
- Entropy As a Function of Temperature & Volume, Cont.31:55
- Volume-dependence of Entropy at Constant Temperature31:56
- Differentiate with Respect to Temperature, Holding Volume Constant36:16
- Recall the Cyclic Rule45:15
- Summary & Recap46:47
- Fundamental Equation of Thermodynamics46:48
- For Entropy as a Function of Temperature & Volume47:18
- The Volume-dependence of Entropy for Liquids & Solids52:52

31m 18s

- Intro0:00
- Entropy as a Function of Temperature & Pressure0:17
- Entropy as a Function of Temperature & Pressure0:18
- Rewrite the Total Differential5:54
- Temperature-dependence7:08
- Pressure-dependence9:04
- Differentiate with Respect to Pressure & Holding Temperature Constant9:54
- Differentiate with Respect to Temperature & Holding Pressure Constant11:28
- Pressure-Dependence of Entropy for Liquids & Solids18:45
- Pressure-Dependence of Entropy for Liquids & Solids18:46
- Example I: ∆S of Transformation26:20

23m 6s

- Intro0:00
- Summary of Entropy So Far0:43
- Defining dS1:04
- Fundamental Equation of Thermodynamics3:51
- Temperature & Volume6:04
- Temperature & Pressure9:10
- Two Important Equations for How Entropy Behaves13:38
- State of a System & Heat Capacity15:34
- Temperature-dependence of Entropy19:49

25m 42s

- Intro0:00
- Entropy Changes for an Ideal Gas1:10
- General Equation1:22
- The Fundamental Theorem of Thermodynamics2:37
- Recall the Basic Total Differential Expression for S = S (T,V)5:36
- For a Finite Change in State7:58
- If Cv is Constant Over the Particular Temperature Range9:05
- Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure11:35
- Change in Entropy of an Ideal Gas as a Function of Temperature & Pressure11:36
- Recall the Basic Total Differential expression for S = S (T, P)15:13
- For a Finite Change18:06
- Example 1: Calculate the ∆S of Transformation22:02

43m 39s

- Intro0:00
- Entropy Example Problems I0:24
- Fundamental Equation of Thermodynamics1:10
- Entropy as a Function of Temperature & Volume2:04
- Entropy as a Function of Temperature & Pressure2:59
- Entropy For Phase Changes4:47
- Entropy For an Ideal Gas6:14
- Third Law Entropies8:25
- Statement of the Third Law9:17
- Entropy of the Liquid State of a Substance Above Its Melting Point10:23
- Entropy For the Gas Above Its Boiling Temperature13:02
- Entropy Changes in Chemical Reactions15:26
- Entropy Change at a Temperature Other than 25°C16:32
- Example I19:31
- Part A: Calculate ∆S for the Transformation Under Constant Volume20:34
- Part B: Calculate ∆S for the Transformation Under Constant Pressure25:04
- Example II: Calculate ∆S fir the Transformation Under Isobaric Conditions27:53
- Example III30:14
- Part A: Calculate ∆S if 1 Mol of Aluminum is taken from 25°C to 255°C31:14
- Part B: If S°₂₉₈ = 28.4 J/mol-K, Calculate S° for Aluminum at 498 K33:23
- Example IV: Calculate Entropy Change of Vaporization for CCl₄34:19
- Example V35:41
- Part A: Calculate ∆S of Transformation37:36
- Part B: Calculate ∆S of Transformation39:10

56m 44s

- Intro0:00
- Example I0:09
- Example I: Calculate ∆U1:28
- Example I: Calculate Q3:29
- Example I: Calculate Cp4:54
- Example I: Calculate ∆S6:14
- Example II7:13
- Example II: Calculate W8:14
- Example II: Calculate ∆U8:56
- Example II: Calculate Q10:18
- Example II: Calculate ∆H11:00
- Example II: Calculate ∆S12:36
- Example III18:47
- Example III: Calculate ∆H19:38
- Example III: Calculate Q21:14
- Example III: Calculate ∆U21:44
- Example III: Calculate W23:59
- Example III: Calculate ∆S24:55
- Example IV27:57
- Example IV: Diagram29:32
- Example IV: Calculate W32:27
- Example IV: Calculate ∆U36:36
- Example IV: Calculate Q38:32
- Example IV: Calculate ∆H39:00
- Example IV: Calculate ∆S40:27
- Example IV: Summary43:41
- Example V48:25
- Example V: Diagram49:05
- Example V: Calculate W50:58
- Example V: Calculate ∆U53:29
- Example V: Calculate Q53:44
- Example V: Calculate ∆H54:34
- Example V: Calculate ∆S55:01

57m 6s

- Intro0:00
- Example I: Isothermal Expansion0:09
- Example I: Calculate W1:19
- Example I: Calculate ∆U1:48
- Example I: Calculate Q2:06
- Example I: Calculate ∆H2:26
- Example I: Calculate ∆S3:02
- Example II: Adiabatic and Reversible Expansion6:10
- Example II: Calculate Q6:48
- Example II: Basic Equation for the Reversible Adiabatic Expansion of an Ideal Gas8:12
- Example II: Finding Volume12:40
- Example II: Finding Temperature17:58
- Example II: Calculate ∆U19:53
- Example II: Calculate W20:59
- Example II: Calculate ∆H21:42
- Example II: Calculate ∆S23:42
- Example III: Calculate the Entropy of Water Vapor25:20
- Example IV: Calculate the Molar ∆S for the Transformation34:32
- Example V44:19
- Part A: Calculate the Standard Entropy of Liquid Lead at 525°C46:17
- Part B: Calculate ∆H for the Transformation of Solid Lead from 25°C to Liquid Lead at 525°C52:23

54m 35s

- Intro0:00
- Entropy & Probability0:11
- Structural Model3:05
- Recall the Fundamental Equation of Thermodynamics9:11
- Two Independent Ways of Affecting the Entropy of a System10:05
- Boltzmann Definition12:10
- Omega16:24
- Definition of Omega16:25
- Energy Distribution19:43
- The Energy Distribution19:44
- In How Many Ways can N Particles be Distributed According to the Energy Distribution23:05
- Example I: In How Many Ways can the Following Distribution be Achieved32:51
- Example II: In How Many Ways can the Following Distribution be Achieved33:51
- Example III: In How Many Ways can the Following Distribution be Achieved34:45
- Example IV: In How Many Ways can the Following Distribution be Achieved38:50
- Entropy & Probability, cont.40:57
- More on Distribution40:58
- Example I Summary41:43
- Example II Summary42:12
- Distribution that Maximizes Omega42:26
- If Omega is Large, then S is Large44:22
- Two Constraints for a System to Achieve the Highest Entropy Possible47:07
- What Happened When the Energy of a System is Increased?49:00

35m 5s

- Intro0:00
- Volume Distribution0:08
- Distributing 2 Balls in 3 Spaces1:43
- Distributing 2 Balls in 4 Spaces3:44
- Distributing 3 Balls in 10 Spaces5:30
- Number of Ways to Distribute P Particles over N Spaces6:05
- When N is Much Larger than the Number of Particles P7:56
- Energy Distribution25:04
- Volume Distribution25:58
- Entropy, Total Entropy, & Total Omega Equations27:34
- Entropy, Total Entropy, & Total Omega Equations27:35

28m 42s

- Intro0:00
- Reversible & Irreversible0:24
- Reversible vs. Irreversible0:58
- Defining Equation for Equilibrium2:11
- Defining Equation for Irreversibility (Spontaneity)3:11
- TdS ≥ dQ5:15
- Transformation in an Isolated System11:22
- Transformation in an Isolated System11:29
- Transformation at Constant Temperature14:50
- Transformation at Constant Temperature14:51
- Helmholtz Free Energy17:26
- Define: A = U - TS17:27
- Spontaneous Isothermal Process & Helmholtz Energy20:20
- Pressure-volume Work22:02

34m 38s

- Intro0:00
- Transformation under Constant Temperature & Pressure0:08
- Transformation under Constant Temperature & Pressure0:36
- Define: G = U + PV - TS3:32
- Gibbs Energy5:14
- What Does This Say?6:44
- Spontaneous Process & a Decrease in G14:12
- Computing ∆G18:54
- Summary of Conditions21:32
- Constraint & Condition for Spontaneity21:36
- Constraint & Condition for Equilibrium24:54
- A Few Words About the Word Spontaneous26:24
- Spontaneous Does Not Mean Fast26:25
- Putting Hydrogen & Oxygen Together in a Flask26:59
- Spontaneous Vs. Not Spontaneous28:14
- Thermodynamically Favorable29:03
- Example: Making a Process Thermodynamically Favorable29:34
- Driving Forces for Spontaneity31:35
- Equation: ∆G = ∆H - T∆S31:36
- Always Spontaneous Process32:39
- Never Spontaneous Process33:06
- A Process That is Endothermic Can Still be Spontaneous34:00

30m 50s

- Intro0:00
- The Fundamental Equations of Thermodynamics0:44
- Mechanical Properties of a System0:45
- Fundamental Properties of a System1:16
- Composite Properties of a System1:44
- General Condition of Equilibrium3:16
- Composite Functions & Their Differentiations6:11
- dH = TdS + VdP7:53
- dA = -SdT - PdV9:26
- dG = -SdT + VdP10:22
- Summary of Equations12:10
- Equation #114:33
- Equation #215:15
- Equation #315:58
- Equation #416:42
- Maxwell's Relations20:20
- Maxwell's Relations20:21
- Isothermal Volume-Dependence of Entropy & Isothermal Pressure-Dependence of Entropy26:21

34m 6s

- Intro0:00
- The General Thermodynamic Equations of State0:10
- Equations of State for Liquids & Solids0:52
- More General Condition for Equilibrium4:02
- General Conditions: Equation that Relates P to Functions of T & V6:20
- The Second Fundamental Equation of Thermodynamics11:10
- Equation 117:34
- Equation 221:58
- Recall the General Expression for Cp - Cv28:11
- For the Joule-Thomson Coefficient30:44
- Joule-Thomson Inversion Temperature32:12

39m 18s

- Intro0:00
- Properties of the Helmholtz & Gibbs Energies0:10
- Equating the Differential Coefficients1:34
- An Increase in T; a Decrease in A3:25
- An Increase in V; a Decrease in A6:04
- We Do the Same Thing for G8:33
- Increase in T; Decrease in G10:50
- Increase in P; Decrease in G11:36
- 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 Constant18:57
- For an Ideal Gas22:18
- Special Note24:56
- Temperature Dependence of Gibbs Energy27:02
- Temperature Dependence of Gibbs Energy #127:52
- Temperature Dependence of Gibbs Energy #229:01
- Temperature Dependence of Gibbs Energy #329:50
- Temperature Dependence of Gibbs Energy #434:50

19m 40s

- Intro0:00
- Entropy of the Universe & the Surroundings0:08
- Equation: ∆G = ∆H - T∆S0:20
- Conditions of Constant Temperature & Pressure1:14
- Reversible Process3:14
- Spontaneous Process & the Entropy of the Universe5:20
- Tips for Remembering Everything12:40
- Verify Using Known Spontaneous Process14:51

54m 16s

- Intro0:00
- Example I0:11
- Example I: Deriving a Function for Entropy (S)2:06
- Example I: Deriving a Function for V5:55
- Example I: Deriving a Function for H8:06
- Example I: Deriving a Function for U12:06
- Example II15:18
- Example III21:52
- Example IV26:12
- Example IV: Part A26:55
- Example IV: Part B28:30
- Example IV: Part C30:25
- Example V33:45
- Example VI40:46
- Example VII43:43
- Example VII: Part A44:46
- Example VII: Part B50:52
- Example VII: Part C51:56

31m 17s

- Intro0:00
- Example I0:09
- Example II5:18
- Example III8:22
- Example IV12:32
- Example V17:14
- Example VI20:34
- Example VI: Part A21:04
- Example VI: Part B23:56
- Example VI: Part C27:56

45m

- Intro0:00
- Example I0:10
- Example II15:03
- Example III21:47
- Example IV28:37
- Example IV: Part A29:33
- Example IV: Part B36:09
- Example IV: Part C40:34

58m 5s

- Intro0:00
- Example I0:41
- Part A: Calculating ∆H3:55
- Part B: Calculating ∆S15:13
- Example II24:39
- Part A: Final Temperature of the System26:25
- Part B: Calculating ∆S36:57
- Example III46:49

25m 20s

- Intro0:00
- Work, Heat, and Energy0:18
- Definition of Work, Energy, Enthalpy, and Heat Capacities0:23
- Heat Capacities for an Ideal Gas3:40
- Path Property & State Property3:56
- Energy Differential5:04
- Enthalpy Differential5:40
- Joule's Law & Joule-Thomson Coefficient6:23
- Coefficient of Thermal Expansion & Coefficient of Compressibility7:01
- Enthalpy of a Substance at Any Other Temperature7:29
- Enthalpy of a Reaction at Any Other Temperature8:01
- Entropy8:53
- Definition of Entropy8:54
- Clausius Inequality9:11
- Entropy Changes in Isothermal Systems9:44
- The Fundamental Equation of Thermodynamics10:12
- Expressing Entropy Changes in Terms of Properties of the System10:42
- Entropy Changes in the Ideal Gas11:22
- Third Law Entropies11:38
- Entropy Changes in Chemical Reactions14:02
- Statistical Definition of Entropy14:34
- Omega for the Spatial & Energy Distribution14:47
- Spontaneity and Equilibrium15:43
- Helmholtz Energy & Gibbs Energy15:44
- Condition for Spontaneity & Equilibrium16:24
- Condition for Spontaneity with Respect to Entropy17:58
- The Fundamental Equations18:30
- Maxwell's Relations19:04
- The Thermodynamic Equations of State20:07
- Energy & Enthalpy Differentials21:08
- Joule's Law & Joule-Thomson Coefficient21:59
- Relationship Between Constant Pressure & Constant Volume Heat Capacities23:14
- One Final Equation - Just for Fun24:04

34m 25s

- Intro0:00
- Complex Numbers0:11
- Representing Complex Numbers in the 2-Dimmensional Plane0:56
- Addition of Complex Numbers2:35
- Subtraction of Complex Numbers3:17
- Multiplication of Complex Numbers3:47
- Division of Complex Numbers6:04
- r & θ8:04
- Euler's Formula11:00
- Polar Exponential Representation of the Complex Numbers11:22
- Example I14:25
- Example II15:21
- Example III16:58
- Example IV18:35
- Example V20:40
- Example VI21:32
- Example VII25:22

59m 57s

- Intro0:00
- Probability & Statistics1:51
- Normalization Condition1:52
- Define the Mean or Average of x11:04
- Example I: Calculate the Mean of x14:57
- Example II: Calculate the Second Moment of the Data in Example I22:39
- Define the Second Central Moment or Variance25:26
- Define the Second Central Moment or Variance25:27
- 1st Term32:16
- 2nd Term32:40
- 3rd Term34:07
- Continuous Distributions35:47
- Continuous Distributions35:48
- Probability Density39:30
- Probability Density39:31
- Normalization Condition46:51
- Example III50:13
- Part A - Show that P(x) is Normalized51:40
- Part B - Calculate the Average Position of the Particle Along the Interval54:31
- Important Things to Remember58:24

42m 5s

- Intro0:00
- Schrӧdinger Equation & Operators0:16
- Relation Between a Photon's Momentum & Its Wavelength0:17
- Louis de Broglie: Wavelength for Matter0:39
- Schrӧdinger Equation1:19
- Definition of Ψ(x)3:31
- Quantum Mechanics5:02
- Operators7:51
- Example I10:10
- Example II11:53
- Example III14:24
- Example IV17:35
- Example V19:59
- Example VI22:39
- Operators Can Be Linear or Non Linear27:58
- Operators Can Be Linear or Non Linear28:34
- Example VII32:47
- Example VIII36:55
- Example IX39:29

30m 26s

- Intro0:00
- Schrӧdinger Equation as an Eigenvalue Problem0:10
- Operator: Multiplying the Original Function by Some Scalar0:11
- Operator, Eigenfunction, & Eigenvalue4:42
- Example: Eigenvalue Problem8:00
- Schrӧdinger Equation as an Eigenvalue Problem9:24
- Hamiltonian Operator15:09
- Quantum Mechanical Operators16:46
- Kinetic Energy Operator19:16
- Potential Energy Operator20:02
- Total Energy Operator21:12
- Classical Point of View21:48
- Linear Momentum Operator24:02
- Example I26:01

21m 34s

- Intro0:00
- The Plausibility of the Schrӧdinger Equation1:16
- The Plausibility of the Schrӧdinger Equation, Part 11:17
- The Plausibility of the Schrӧdinger Equation, Part 28:24
- The Plausibility of the Schrӧdinger Equation, Part 313:45

56m 22s

- Intro0:00
- Free Particle in a Box0:28
- Definition of a Free Particle in a Box0:29
- Amplitude of the Matter Wave6:22
- Intensity of the Wave6:53
- Probability Density9:39
- Probability that the Particle is Located Between x & dx10:54
- Probability that the Particle will be Found Between o & a12:35
- Wave Function & the Particle14:59
- Boundary Conditions19:22
- What Happened When There is No Constraint on the Particle27:54
- Diagrams34:12
- More on Probability Density40:53
- The Correspondence Principle46:45
- The Correspondence Principle46:46
- Normalizing the Wave Function47:46
- Normalizing the Wave Function47:47
- Normalized Wave Function & Normalization Constant52:24

45m 24s

- Intro0:00
- Free Particle in a Box0:08
- Free Particle in a 1-dimensional Box0:09
- For a Particle in a Box3:57
- Calculating Average Values & Standard Deviations5:42
- Average Value for the Position of a Particle6:32
- Standard Deviations for the Position of a Particle10:51
- Recall: Energy & Momentum are Represented by Operators13:33
- Recall: Schrӧdinger Equation in Operator Form15:57
- Average Value of a Physical Quantity that is Associated with an Operator18:16
- Average Momentum of a Free Particle in a Box20:48
- The Uncertainty Principle24:42
- Finding the Standard Deviation of the Momentum25:08
- Expression for the Uncertainty Principle35:02
- Summary of the Uncertainty Principle41:28

48m 43s

- Intro0:00
- 2-Dimension0:12
- Dimension 20:31
- Boundary Conditions1:52
- Partial Derivatives4:27
- Example I6:08
- The Particle in a Box, cont.11:28
- Operator Notation12:04
- Symbol for the Laplacian13:50
- The Equation Becomes…14:30
- Boundary Conditions14:54
- Separation of Variables15:33
- Solution to the 1-dimensional Case16:31
- Normalization Constant22:32
- 3-Dimension28:30
- Particle in a 3-dimensional Box28:31
- In Del Notation32:22
- The Solutions34:51
- Expressing the State of the System for a Particle in a 3D Box39:10
- Energy Level & Degeneracy43:35

46m 18s

- Intro0:00
- Postulate I0:31
- Probability That The Particle Will Be Found in a Differential Volume Element0:32
- Example I: Normalize This Wave Function11:30
- Postulate II18:20
- Postulate II18:21
- Quantum Mechanical Operators: Position20:48
- Quantum Mechanical Operators: Kinetic Energy21:57
- Quantum Mechanical Operators: Potential Energy22:42
- Quantum Mechanical Operators: Total Energy22:57
- Quantum Mechanical Operators: Momentum23:22
- Quantum Mechanical Operators: Angular Momentum23:48
- More On The Kinetic Energy Operator24:48
- Angular Momentum28:08
- Angular Momentum Overview28:09
- Angular Momentum Operator in Quantum Mechanic31:34
- The Classical Mechanical Observable32:56
- Quantum Mechanical Operator37:01
- Getting the Quantum Mechanical Operator from the Classical Mechanical Observable40:16
- Postulate II, cont.43:40
- Quantum Mechanical Operators are Both Linear & Hermetical43:41

39m 28s

- Intro0:00
- Postulate III0:09
- Postulate III: Part I0:10
- Postulate III: Part II5:56
- Postulate III: Part III12:43
- Postulate III: Part IV18:28
- Postulate IV23:57
- Postulate IV23:58
- Postulate V27:02
- Postulate V27:03
- Average Value36:38
- Average Value36:39

35m 32s

- Intro0:00
- The Postulates & Principles of Quantum Mechanics, Part III0:10
- Equations: Linear & Hermitian0:11
- Introduction to Hermitian Property3:36
- Eigenfunctions are Orthogonal9:55
- The Sequence of Wave Functions for the Particle in a Box forms an Orthonormal Set14:34
- Definition of Orthogonality16:42
- Definition of Hermiticity17:26
- Hermiticity: The Left Integral23:04
- Hermiticity: The Right Integral28:47
- Hermiticity: Summary34:06

29m 55s

- Intro0:00
- The Postulates & Principles of Quantum Mechanics, Part IV0:09
- Operators can be Applied Sequentially0:10
- Sample Calculation 12:41
- Sample Calculation 25:18
- Commutator of Two Operators8:16
- The Uncertainty Principle19:01
- In the Case of Linear Momentum and Position Operator23:14
- When the Commutator of Two Operators Equals to Zero26:31

54m 25s

- Intro0:00
- Example I: Three Dimensional Box & Eigenfunction of The Laplacian Operator0:37
- Example II: Positions of a Particle in a 1-dimensional Box15:46
- Example III: Transition State & Frequency29:29
- Example IV: Finding a Particle in a 1-dimensional Box35:03
- Example V: Degeneracy & Energy Levels of a Particle in a Box44:59

46m 58s

- Intro0:00
- Review0:25
- Wave Function0:26
- Normalization Condition2:28
- Observable in Classical Mechanics & Linear/Hermitian Operator in Quantum Mechanics3:36
- Hermitian6:11
- Eigenfunctions & Eigenvalue8:20
- Normalized Wave Functions12:00
- Average Value13:42
- If Ψ is Written as a Linear Combination15:44
- Commutator16:45
- Example I: Normalize The Wave Function19:18
- Example II: Probability of Finding of a Particle22:27
- Example III: Orthogonal26:00
- Example IV: Average Value of the Kinetic Energy Operator30:22
- Example V: Evaluate These Commutators39:02

44m 11s

- Intro0:00
- Example I: Good Candidate for a Wave Function0:08
- Example II: Variance of the Energy7:00
- Example III: Evaluate the Angular Momentum Operators15:00
- Example IV: Real Eigenvalues Imposes the Hermitian Property on Operators28:44
- Example V: A Demonstration of Why the Eigenfunctions of Hermitian Operators are Orthogonal35:33

35m 33s

- Intro0:00
- The Harmonic Oscillator0:10
- Harmonic Motion0:11
- Classical Harmonic Oscillator4:38
- Hooke's Law8:18
- Classical Harmonic Oscillator, cont.10:33
- General Solution for the Differential Equation15:16
- Initial Position & Velocity16:05
- Period & Amplitude20:42
- Potential Energy of the Harmonic Oscillator23:20
- Kinetic Energy of the Harmonic Oscillator26:37
- Total Energy of the Harmonic Oscillator27:23
- Conservative System34:37

43m 4s

- Intro0:00
- The Harmonic Oscillator II0:08
- Diatomic Molecule0:10
- Notion of Reduced Mass5:27
- Harmonic Oscillator Potential & The Intermolecular Potential of a Vibrating Molecule7:33
- The Schrӧdinger Equation for the 1-dimensional Quantum Mechanic Oscillator14:14
- Quantized Values for the Energy Level15:46
- Ground State & the Zero-Point Energy21:50
- Vibrational Energy Levels25:18
- Transition from One Energy Level to the Next26:42
- Fundamental Vibrational Frequency for Diatomic Molecule34:57
- Example: Calculate k38:01

26m 30s

- Intro0:00
- The Harmonic Oscillator III0:09
- The Wave Functions Corresponding to the Energies0:10
- Normalization Constant2:34
- Hermite Polynomials3:22
- First Few Hermite Polynomials4:56
- First Few Wave-Functions6:37
- Plotting the Probability Density of the Wave-Functions8:37
- Probability Density for Large Values of r14:24
- Recall: Odd Function & Even Function19:05
- More on the Hermite Polynomials20:07
- Recall: If f(x) is Odd20:36
- Average Value of x22:31
- Average Value of Momentum23:56

41m 10s

- Intro0:00
- Possible Confusion from the Previous Discussion0:07
- Possible Confusion from the Previous Discussion0:08
- Rotation of a Single Mass Around a Fixed Center8:17
- Rotation of a Single Mass Around a Fixed Center8:18
- Angular Velocity12:07
- Rotational Inertia13:24
- Rotational Frequency15:24
- Kinetic Energy for a Linear System16:38
- Kinetic Energy for a Rotational System17:42
- Rotating Diatomic Molecule19:40
- Rotating Diatomic Molecule: Part 119:41
- Rotating Diatomic Molecule: Part 224:56
- Rotating Diatomic Molecule: Part 330:04
- Hamiltonian of the Rigid Rotor36:48
- Hamiltonian of the Rigid Rotor36:49

30m 32s

- Intro0:00
- The Rigid Rotator II0:08
- Cartesian Coordinates0:09
- Spherical Coordinates1:55
- r6:15
- θ6:28
- φ7:00
- Moving a Distance 'r'8:17
- Moving a Distance 'r' in the Spherical Coordinates11:49
- For a Rigid Rotator, r is Constant13:57
- Hamiltonian Operator15:09
- Square of the Angular Momentum Operator17:34
- Orientation of the Rotation in Space19:44
- Wave Functions for the Rigid Rotator20:40
- The Schrӧdinger Equation for the Quantum Mechanic Rigid Rotator21:24
- Energy Levels for the Rigid Rotator26:58

35m 19s

- Intro0:00
- The Rigid Rotator III0:11
- When a Rotator is Subjected to Electromagnetic Radiation1:24
- Selection Rule2:13
- Frequencies at Which Absorption Transitions Occur6:24
- Energy Absorption & Transition10:54
- Energy of the Individual Levels Overview20:58
- Energy of the Individual Levels: Diagram23:45
- Frequency Required to Go from J to J + 125:53
- Using Separation Between Lines on the Spectrum to Calculate Bond Length28:02
- Example I: Calculating Rotational Inertia & Bond Length29:18
- Example I: Calculating Rotational Inertia29:19
- Example I: Calculating Bond Length32:56

33m 48s

- Intro0:00
- Equations Review0:11
- Energy of the Harmonic Oscillator0:12
- Selection Rule3:02
- Observed Frequency of Radiation3:27
- Harmonic Oscillator Wave Functions5:52
- Rigid Rotator7:26
- Selection Rule for Rigid Rotator9:15
- Frequency of Absorption9:35
- Wave Numbers10:58
- Example I: Calculate the Reduced Mass of the Hydrogen Atom11:44
- Example II: Calculate the Fundamental Vibration Frequency & the Zero-Point Energy of This Molecule13:37
- Example III: Show That the Product of Two Even Functions is even19:35
- Example IV: Harmonic Oscillator24:56

46m 43s

- Intro0:00
- Example I: Harmonic Oscillator0:12
- Example II: Harmonic Oscillator23:26
- Example III: Calculate the RMS Displacement of the Molecules38:12

40m

- Intro0:00
- The Hydrogen Atom I1:31
- Review of the Rigid Rotator1:32
- Hydrogen Atom & the Coulomb Potential2:50
- Using the Spherical Coordinates6:33
- Applying This Last Expression to Equation 110:19
- Angular Component & Radial Component13:26
- Angular Equation15:56
- Solution for F(φ)19:32
- Determine The Normalization Constant20:33
- Differential Equation for T(a)24:44
- Legendre Equation27:20
- Legendre Polynomials31:20
- The Legendre Polynomials are Mutually Orthogonal35:40
- Limits37:17
- Coefficients38:28

35m 58s

- Intro0:00
- Associated Legendre Functions0:07
- Associated Legendre Functions0:08
- First Few Associated Legendre Functions6:39
- s, p, & d Orbital13:24
- The Normalization Condition15:44
- Spherical Harmonics20:03
- Equations We Have Found20:04
- Wave Functions for the Angular Component & Rigid Rotator24:36
- Spherical Harmonics Examples25:40
- Angular Momentum30:09
- Angular Momentum30:10
- Square of the Angular Momentum35:38
- Energies of the Rigid Rotator38:21

36m 18s

- Intro0:00
- The Hydrogen Atom III0:34
- Angular Momentum is a Vector Quantity0:35
- The Operators Corresponding to the Three Components of Angular Momentum Operator: In Cartesian Coordinates1:30
- The Operators Corresponding to the Three Components of Angular Momentum Operator: In Spherical Coordinates3:27
- Z Component of the Angular Momentum Operator & the Spherical Harmonic5:28
- Magnitude of the Angular Momentum Vector20:10
- Classical Interpretation of Angular Momentum25:22
- Projection of the Angular Momentum Vector onto the xy-plane33:24

33m 55s

- Intro0:00
- The Hydrogen Atom IV0:09
- The Equation to Find R( r )0:10
- Relation Between n & l3:50
- The Solutions for the Radial Functions5:08
- Associated Laguerre Polynomials7:58
- 1st Few Associated Laguerre Polynomials8:55
- Complete Wave Function for the Atomic Orbitals of the Hydrogen Atom12:24
- The Normalization Condition15:06
- In Cartesian Coordinates18:10
- Working in Polar Coordinates20:48
- Principal Quantum Number21:58
- Angular Momentum Quantum Number22:35
- Magnetic Quantum Number25:55
- Zeeman Effect30:45

51m 53s

- Intro0:00
- The Hydrogen Atom V: Where We Are0:13
- Review0:14
- Let's Write Out ψ₂₁₁7:32
- Angular Momentum of the Electron14:52
- Representation of the Wave Function19:36
- Radial Component28:02
- Example: 1s Orbital28:34
- Probability for Radial Function33:46
- 1s Orbital: Plotting Probability Densities vs. r35:47
- 2s Orbital: Plotting Probability Densities vs. r37:46
- 3s Orbital: Plotting Probability Densities vs. r38:49
- 4s Orbital: Plotting Probability Densities vs. r39:34
- 2p Orbital: Plotting Probability Densities vs. r40:12
- 3p Orbital: Plotting Probability Densities vs. r41:02
- 4p Orbital: Plotting Probability Densities vs. r41:51
- 3d Orbital: Plotting Probability Densities vs. r43:18
- 4d Orbital: Plotting Probability Densities vs. r43:48
- Example I: Probability of Finding an Electron in the 2s Orbital of the Hydrogen45:40

51m 53s

- Intro0:00
- The Hydrogen Atom VI0:07
- Last Lesson Review0:08
- Spherical Component1:09
- Normalization Condition2:02
- Complete 1s Orbital Wave Function4:08
- 1s Orbital Wave Function4:09
- Normalization Condition6:28
- Spherically Symmetric16:00
- Average Value17:52
- Example I: Calculate the Region of Highest Probability for Finding the Electron21:19
- 2s Orbital Wave Function25:32
- 2s Orbital Wave Function25:33
- Average Value28:56
- General Formula32:24

34m 29s

- Intro0:00
- The Hydrogen Atom VII0:12
- p Orbitals1:30
- Not Spherically Symmetric5:10
- Recall That the Spherical Harmonics are Eigenfunctions of the Hamiltonian Operator6:50
- Any Linear Combination of These Orbitals Also Has The Same Energy9:16
- Functions of Real Variables15:53
- Solving for Px16:50
- Real Spherical Harmonics21:56
- Number of Nodes32:56

43m 49s

- Intro0:00
- Example I: Angular Momentum & Spherical Harmonics0:20
- Example II: Pair-wise Orthogonal Legendre Polynomials16:40
- Example III: General Normalization Condition for the Legendre Polynomials25:06
- Example IV: Associated Legendre Functions32:13

1h 1m 57s

- Intro0:00
- Example I: Normalization & Pair-wise Orthogonal0:13
- Part 1: Normalized0:43
- Part 2: Pair-wise Orthogonal16:53
- Example II: Show Explicitly That the Following Statement is True for Any Integer n27:10
- Example III: Spherical Harmonics29:26
- Angular Momentum Cones56:37
- Angular Momentum Cones56:38
- Physical Interpretation of Orbital Angular Momentum in Quantum mechanics1:00:16

48m 33s

- Intro0:00
- Example I: Show That ψ₂₁₁ is Normalized0:07
- Example II: Show That ψ₂₁₁ is Orthogonal to ψ₃₁₀11:48
- Example III: Probability That a 1s Electron Will Be Found Within 1 Bohr Radius of The Nucleus18:35
- Example IV: Radius of a Sphere26:06
- Example V: Calculate <r> for the 2s Orbital of the Hydrogen-like Atom36:33

48m 33s

- Intro0:00
- Example I: Probability Density vs. Radius Plot0:11
- Example II: Hydrogen Atom & The Coulombic Potential14:16
- Example III: Find a Relation Among <K>, <V>, & <E>25:47
- Example IV: Quantum Mechanical Virial Theorem48:32
- Example V: Find the Variance for the 2s Orbital54:13

48m 33s

- Intro0:00
- Example I: Derive a Formula for the Degeneracy of a Given Level n0:11
- Example II: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ8:30
- Example III: Using Linear Combinations to Represent the Spherical Harmonics as Functions of the Real Variables θ & φ23:01
- Example IV: Orbital Functions31:51

59m 18s

- Intro0:00
- Quantum Numbers Specify an Orbital0:24
- n1:10
- l1:20
- m1:35
- 4th Quantum Number: s2:02
- Spin Orbitals7:03
- Spin Orbitals7:04
- Multi-electron Atoms11:08
- Term Symbols18:08
- Russell-Saunders Coupling & The Atomic Term Symbol18:09
- Example: Configuration for C27:50
- Configuration for C: 1s²2s²2p²27:51
- Drawing Every Possible Arrangement31:15
- Term Symbols45:24
- Microstate50:54

34m 54s

- Intro0:00
- Microstates0:25
- We Started With 21 Possible Microstates0:26
- ³P State2:05
- Microstates in ³P Level5:10
- ¹D State13:16
- ³P State16:10
- ²P₂ State17:34
- ³P₁ State18:34
- ³P₀ State19:12
- 9 Microstates in ³P are Subdivided19:40
- ¹S State21:44
- Quicker Way to Find the Different Values of J for a Given Basic Term Symbol22:22
- Ground State26:27
- Hund's Empirical Rules for Specifying the Term Symbol for the Ground Electronic State27:29
- Hund's Empirical Rules: 128:24
- Hund's Empirical Rules: 229:22
- Hund's Empirical Rules: 3 - Part A30:22
- Hund's Empirical Rules: 3 - Part B31:18
- Example: 1s²2s²2p²31:54

38m 3s

- Intro0:00
- Spin Quantum Number: Term Symbols III0:14
- Deriving the Term Symbols for the p² Configuration0:15
- Table: MS vs. ML3:57
- ¹D State16:21
- ³P State21:13
- ¹S State24:48
- J Value25:32
- Degeneracy of the Level27:28
- When Given r Electrons to Assign to n Equivalent Spin Orbitals30:18
- p² Configuration32:51
- Complementary Configurations35:12

57m 49s

- Intro0:00
- Lyman Series0:09
- Spectroscopic Term Symbols0:10
- Lyman Series3:04
- Hydrogen Levels8:21
- Hydrogen Levels8:22
- Term Symbols & Atomic Spectra14:17
- Spin-Orbit Coupling14:18
- Selection Rules for Atomic Spectra21:31
- Selection Rules for Possible Transitions23:56
- Wave Numbers for The Transitions28:04
- Example I: Calculate the Frequencies of the Allowed Transitions from (4d) ²D →(2p) ²P32:23
- Helium Levels49:50
- Energy Levels for Helium49:51
- Transitions & Spin Multiplicity52:27
- Transitions & Spin Multiplicity52:28

1h 1m 20s

- Intro0:00
- 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

56m 34s

- Intro0:00
- Example I: Find the Term Symbols for the nd² Configuration0:11
- Example II: Find the Term Symbols for the 1s¹2p¹ Configuration27:02
- Example III: Calculate the Separation Between the Doublets in the Lyman Series for Atomic Hydrogen41:41
- Example IV: Calculate the Frequencies of the Lines for the (4d) ²D → (3p) ²P Transition48:53

18m 24s

- Intro0:00
- Quantum Mechanics Equations0:37
- De Broglie Relation0:38
- Statistical Relations1:00
- The Schrӧdinger Equation1:50
- The Particle in a 1-Dimensional Box of Length a3:09
- The Particle in a 2-Dimensional Box of Area a x b3:48
- The Particle in a 3-Dimensional Box of Area a x b x c4:22
- The Schrӧdinger Equation Postulates4:51
- The Normalization Condition5:40
- The Probability Density6:51
- Linear7:47
- Hermitian8:31
- Eigenvalues & Eigenfunctions8:55
- The Average Value9:29
- Eigenfunctions of Quantum Mechanics Operators are Orthogonal10:53
- Commutator of Two Operators10:56
- The Uncertainty Principle11:41
- The Harmonic Oscillator13:18
- The Rigid Rotator13:52
- Energy of the Hydrogen Atom14:30
- Wavefunctions, Radial Component, and Associated Laguerre Polynomial14:44
- Angular Component or Spherical Harmonic15:16
- Associated Legendre Function15:31
- Principal Quantum Number15:43
- Angular Momentum Quantum Number15:50
- Magnetic Quantum Number16:21
- z-component of the Angular Momentum of the Electron16:53
- Atomic Spectroscopy: Term Symbols17:14
- Atomic Spectroscopy: Selection Rules18:03

50m 2s

- Intro0:00
- Spectroscopic Overview: Which Equation Do I Use & Why1:02
- Lesson Overview1:03
- Rotational & Vibrational Spectroscopy4:01
- Frequency of Absorption/Emission6:04
- Wavenumbers in Spectroscopy8:10
- Starting State vs. Excited State10:10
- Total Energy of a Molecule (Leaving out the Electronic Energy)14:02
- Energy of Rotation: Rigid Rotor15:55
- Energy of Vibration: Harmonic Oscillator19:08
- Equation of the Spectral Lines23:22
- Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)28:37
- Harmonic Oscillator-Rigid Rotor Approximation (Making Corrections)28:38
- Vibration-Rotation Interaction33:46
- Centrifugal Distortion36:27
- Anharmonicity38:28
- Correcting for All Three Simultaneously41:03
- Spectroscopic Parameters44:26
- Summary47:32
- Harmonic Oscillator-Rigid Rotor Approximation47:33
- Vibration-Rotation Interaction48:14
- Centrifugal Distortion48:20
- Anharmonicity48:28
- Correcting for All Three Simultaneously48:44

59m 47s

- Intro0:00
- Vibration-Rotation0:37
- What is Molecular Spectroscopy?0:38
- Microwave, Infrared Radiation, Visible & Ultraviolet1:53
- Equation for the Frequency of the Absorbed Radiation4:54
- Wavenumbers6:15
- Diatomic Molecules: Energy of the Harmonic Oscillator8:32
- Selection Rules for Vibrational Transitions10:35
- Energy of the Rigid Rotator16:29
- Angular Momentum of the Rotator21:38
- Rotational Term F(J)26:30
- Selection Rules for Rotational Transition29:30
- Vibration Level & Rotational States33:20
- Selection Rules for Vibration-Rotation37:42
- Frequency of Absorption39:32
- Diagram: Energy Transition45:55
- Vibration-Rotation Spectrum: HCl51:27
- Vibration-Rotation Spectrum: Carbon Monoxide54:30

46m 22s

- Intro0:00
- Vibration-Rotation Interaction0:13
- Vibration-Rotation Spectrum: HCl0:14
- Bond Length & Vibrational State4:23
- Vibration Rotation Interaction10:18
- Case 112:06
- Case 217:17
- Example I: HCl Vibration-Rotation Spectrum22:58
- Rotational Constant for the 0 & 1 Vibrational State26:30
- Equilibrium Bond Length for the 1 Vibrational State39:42
- Equilibrium Bond Length for the 0 Vibrational State42:13
- Bₑ & αₑ44:54

29m 24s

- Intro0:00
- The Non-Rigid Rotator0:09
- Pure Rotational Spectrum0:54
- The Selection Rules for Rotation3:09
- Spacing in the Spectrum5:04
- Centrifugal Distortion Constant9:00
- Fundamental Vibration Frequency11:46
- Observed Frequencies of Absorption14:14
- Difference between the Rigid Rotator & the Adjusted Rigid Rotator16:51
- Adjusted Rigid Rotator21:31
- Observed Frequencies of Absorption26:26

30m 53s

- Intro0:00
- The Anharmonic Oscillator0:09
- Vibration-Rotation Interaction & Centrifugal Distortion0:10
- Making Corrections to the Harmonic Oscillator4:50
- Selection Rule for the Harmonic Oscillator7:50
- Overtones8:40
- True Oscillator11:46
- Harmonic Oscillator Energies13:16
- Anharmonic Oscillator Energies13:33
- Observed Frequencies of the Overtones15:09
- True Potential17:22
- HCl Vibrational Frequencies: Fundamental & First Few Overtones21:10
- Example I: Vibrational States & Overtones of the Vibrational Spectrum22:42
- Example I: Part A - First 4 Vibrational States23:44
- Example I: Part B - Fundamental & First 3 Overtones25:31
- Important Equations27:45
- Energy of the Q State29:14
- The Difference in Energy between 2 Successive States29:23
- Difference in Energy between 2 Spectral Lines29:40

1h 1m 33s

- Intro0:00
- Electronic Transitions0:16
- Electronic State & Transition0:17
- Total Energy of the Diatomic Molecule3:34
- Vibronic Transitions4:30
- Selection Rule for Vibronic Transitions9:11
- More on Vibronic Transitions10:08
- Frequencies in the Spectrum16:46
- Difference of the Minima of the 2 Potential Curves24:48
- Anharmonic Zero-point Vibrational Energies of the 2 States26:24
- Frequency of the 0 → 0 Vibronic Transition27:54
- Making the Equation More Compact29:34
- Spectroscopic Parameters32:11
- Franck-Condon Principle34:32
- Example I: Find the Values of the Spectroscopic Parameters for the Upper Excited State47:27
- Table of Electronic States and Parameters56:41

33m 47s

- Intro0:00
- Example I: Calculate the Bond Length0:10
- Example II: Calculate the Rotational Constant7:39
- Example III: Calculate the Number of Rotations10:54
- Example IV: What is the Force Constant & Period of Vibration?16:31
- Example V: Part A - Calculate the Fundamental Vibration Frequency21:42
- Example V: Part B - Calculate the Energies of the First Three Vibrational Levels24:12
- Example VI: Calculate the Frequencies of the First 2 Lines of the R & P Branches of the Vib-Rot Spectrum of HBr26:28

1h 1m 5s

- Intro0:00
- Example I: Calculate the Frequencies of the Transitions0:09
- Example II: Specify Which Transitions are Allowed & Calculate the Frequencies of These Transitions22:07
- Example III: Calculate the Vibrational State & Equilibrium Bond Length34:31
- Example IV: Frequencies of the Overtones49:28
- Example V: Vib-Rot Interaction, Centrifugal Distortion, & Anharmonicity54:47

33m 31s

- Intro0:00
- Example I: Part A - Derive an Expression for ∆G( r )0:10
- Example I: Part B - Maximum Vibrational Quantum Number6:10
- Example II: Part A - Derive an Expression for the Dissociation Energy of the Molecule8:29
- Example II: Part B - Equation for ∆G( r )14:00
- Example III: How Many Vibrational States are There for Br₂ before the Molecule Dissociates18:16
- Example IV: Find the Difference between the Two Minima of the Potential Energy Curves20:57
- Example V: Rotational Spectrum30:51

1h 1m 15s

- Intro0:00
- Statistical Thermodynamics: The Big Picture0:10
- Our Big Picture Goal0:11
- Partition Function (Q)2:42
- The Molecular Partition Function (q)4:00
- Consider a System of N Particles6:54
- Ensemble13:22
- Energy Distribution Table15:36
- Probability of Finding a System with Energy16:51
- The Partition Function21:10
- Microstate28:10
- Entropy of the Ensemble30:34
- Entropy of the System31:48
- Expressing the Thermodynamic Functions in Terms of The Partition Function39:21
- The Partition Function39:22
- Pi & U41:20
- Entropy of the System44:14
- Helmholtz Energy48:15
- Pressure of the System49:32
- Enthalpy of the System51:46
- Gibbs Free Energy52:56
- Heat Capacity54:30
- Expressing Q in Terms of the Molecular Partition Function (q)59:31
- Indistinguishable Particles1:02:16
- N is the Number of Particles in the System1:03:27
- The Molecular Partition Function1:05:06
- Quantum States & Degeneracy1:07:46
- Thermo Property in Terms of ln Q1:10:09
- Example: Thermo Property in Terms of ln Q1:13:23

47m 23s

- Intro0:00
- Lesson Overview0:19
- Monatomic Ideal Gases6:40
- Monatomic Ideal Gases Overview6:42
- Finding the Parition Function of Translation8:17
- Finding the Parition Function of Electronics13:29
- Example: Na17:42
- Example: F23:12
- Energy Difference between the Ground State & the 1st Excited State29:27
- The Various Partition Functions for Monatomic Ideal Gases32:20
- Finding P43:16
- Going Back to U = (3/2) RT46:20

54m 9s

- Intro0:00
- Diatomic Gases0:16
- Diatomic Gases0:17
- Zero-Energy Mark for Rotation2:26
- Zero-Energy Mark for Vibration3:21
- Zero-Energy Mark for Electronic5:54
- Vibration Partition Function9:48
- When Temperature is Very Low14:00
- When Temperature is Very High15:22
- Vibrational Component18:48
- Fraction of Molecules in the r Vibration State21:00
- Example: Fraction of Molecules in the r Vib. State23:29
- Rotation Partition Function26:06
- Heteronuclear & Homonuclear Diatomics33:13
- Energy & Heat Capacity36:01
- Fraction of Molecules in the J Rotational Level39:20
- Example: Fraction of Molecules in the J Rotational Level40:32
- Finding the Most Populated Level44:07
- Putting It All Together46:06
- Putting It All Together46:07
- Energy of Translation51:51
- Energy of Rotation52:19
- Energy of Vibration52:42
- Electronic Energy53:35

48m 32s

- Intro0:00
- Example I: Calculate the Fraction of Potassium Atoms in the First Excited Electronic State0:10
- Example II: Show That Each Translational Degree of Freedom Contributes R/2 to the Molar Heat Capacity14:46
- Example III: Calculate the Dissociation Energy21:23
- Example IV: Calculate the Vibrational Contribution to the Molar heat Capacity of Oxygen Gas at 500 K25:46
- Example V: Upper & Lower Quantum State32:55
- Example VI: Calculate the Relative Populations of the J=2 and J=1 Rotational States of the CO Molecule at 25°C42:21

57m 30s

- Intro0:00
- Example I: Make a Plot of the Fraction of CO Molecules in Various Rotational Levels0:10
- Example II: Calculate the Ratio of the Translational Partition Function for Cl₂ and Br₂ at Equal Volume & Temperature8:05
- Example III: Vibrational Degree of Freedom & Vibrational Molar Heat Capacity11:59
- Example IV: Calculate the Characteristic Vibrational & Rotational temperatures for Each DOF45:03

For more information, please see full course syllabus of Physical Chemistry

# Physical Chemistry Schrӧdinger Equation as an Eigenvalue Problem

In this lesson our instructor talks about Schrӧdinger Equation as an Eigenvalue Problem. He also discusses kinetic, potential, and total energy operator. He does one complete example problem to round up the lesson.

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1 answer

Last reply by: Professor Hovasapian

Tue Nov 20, 2018 4:11 AM

Post by Kimberly Davis on November 20, 2018

Hi professor Hovasapian,

The book I'm using is Atkin and they defined the Linear Momentum Operator a bit different. They use px=h/i d/dx however yours is -i*h d/dx. I also put a reference link here as : http://www.eng.fsu.edu/~dommelen/quantum/style_a/ops.html. My question is: is the two equivalent and if so, how? thank you professor so much.

1 answer

Last reply by: Professor Hovasapian

Mon Feb 23, 2015 6:43 PM

Post by Carly Sisk on February 23, 2015

Hey Professor Raffi!

Just to clarify what Anhtuan was saying; the final answer should be positive (hbar^2 * sinx), correct?

1 answer

Last reply by: Professor Hovasapian

Wed Jan 28, 2015 12:32 PM

Post by Anhtuan Tran on January 28, 2015

Hi Professor Hovasapian,

On your last example, when you were doing P^2f, your second last step was i^2 * hbar^2 (-sinx). So the answer should be hbar^2 * sinx, because you already took into account the minus sign from sinx and minus sign from i^2. Therefore, there should be no i^2 left at the end.

Could you please double check it? Thank you.

2 answers

Last reply by: Professor Hovasapian

Sat Sep 27, 2014 6:27 AM

Post by Raj Singh on September 26, 2014

Where did the i in (-iHbar d/dx) came from when deriving linear momentum operator? additionally if you multiply 2 -i wouldnt you get positive i or positive 1?