Unit -1 Quantum Mechanics-Final

Page 1: Course Introduction

• Engineering Physics II (SUBJECT CODE: 303192102)

• Instructors: Dr. Swagata Roy & Dr. Mudra Jadav

• Department of Applied Science and Humanities, PIET, Parul University

Page 2: Unit Overview

• Unit 1: Modern Physics

• Slogan: योग: कर्मसु कौशलम्

• Parul University Digital Learning Content

Page 3: Key Topics in Modern Physics

1. Failures of Classical Mechanics:

   • Stability/Structure of Atom

   • Blackbody Radiation

   • Photoelectric Effect

2. Compton Scattering

3. Wave-Particle Duality

4. Heisenberg's Uncertainty Principle

5. Wave Function and Its Physical Significance

6. Operators and Eigen Functions

7. Energy and Momentum Operators

8. Schrödinger's Wave Equations

   • Time Dependent

   • Time Independent

9. Particle in a 1-D Infinite Well

10. Quantum Tunneling

11. Numerical Problems for Study

Page 4: Classical Mechanics

• Definition: Branch of physics dealing with macroscopic object motion and the forces affecting them.

• Key Concepts:

  • Motion of large objects (>1 micron) explained using classical physics.

  • Fundamental Laws:i. Newton's laws of motionii. Newton’s Inverse Square Law of Gravitationiii. Coulomb’s Inverse Square Law for charged bodiesiv. Lorentz force law for moving charges

• Assumption of classical mechanics: objects can be measured in terms of position, mass, velocity, and acceleration simultaneously and accurately.

• Notable Figure: Sir Isaac Newton (1643–1727)

Page 5: Domains of Mechanics

• Speed Comparison:

  • Classical Mechanics:

    • Applicable speed: Far less than 3 x 10⁸ m/s

  • Relativistic Mechanics: Comparable to speed of light

  • Quantum Mechanics: Applies at atomic scale (<10⁻⁹ m)

Page 6: Introduction to Quantum Mechanics

• Definition: Fundamental theory describing matter and energy behavior at small scales (atomic/sub-atomic).

• Key Idea: Probabilistic nature of quantum mechanics; position and momentum of particles cannot be measured simultaneously.

Page 7: Insights into Quantum Mechanics

• Max Planck: Developed quantum theory to explain blackbody radiation.

• Einstein: Utilized quantum theory for the photoelectric effect; established mass-energy equivalence (E = mc²).

• Louis de Broglie: Introduced wave-particle duality, suggesting all matter has wave characteristics.

Page 8: Pioneers of Quantum Mechanics

• Timeline Events:

  • 1900: Blackbody radiation

  • 1905: Photoelectric effect

  • 1911: Atomic structure model

  • 1924: De Broglie wave-particle duality proposal

  • 1927: Heisenberg Uncertainty Principle

  • 1928: Development of relativistic quantum mechanics

  • 1948-1950: Advancement in quantum electrodynamics

Page 9: Failures of Classical Mechanics

• Classical Mechanics: Describes macroscopic object motion; effective for large, relatively slow objects.

• Failures: Explained phenomena that classical physics couldn't account for:

  1. Stability and structure of atoms.

  2. Blackbody radiation results.

  3. Photoelectric effect observations.

Page 10: Atomic Structure and Models

• Historical Models:

  • Thomson’s Watermelon Model

  • Rutherford’s Atom Model

  • Bohr’s Atomic Model: proposed quantized orbits for electrons in atoms.

• Quantum Mechanical Model: Introduced electron clouds (orbitals) instead of fixed paths.

Page 11: Bohr’s Old Quantum Theory

• Stability of Atom:

  • Electrons orbiting nucleus without losing energy or collapsing.

  • Energy quantization: Angular momentum of electrons in stable orbits is quantized.

• Energy Transfer: Electrons can emit/absorb energy only when transitioning between quantized orbits.

Page 12: Introduction to Blackbody Radiation

• Black Body: An ideal body that absorbs all radiation incident on it, exhibiting thermal equilibrium.

• Design of Black Body: Typically modeled as a closed sphere reflecting all incident light.

• Black Body Radiation: Emits radiation across possible wavelengths when heated.

Page 13: Stefan-Boltzmann Law for Blackbody Radiation

• Measurement: Experimental verification by Lummer and Pringsheim (1897).

• Relation: Energy emitted approximately varies as T⁴ (Stefan-Boltzmann Law).

• Formula: E = σT⁴ (σ = Stefan-Boltzmann constant).

Page 14: Spectral Distribution of Blackbody Radiation

• Energy Distribution Characteristics:

  1. Non-uniform distribution across the spectrum at constant temperature.

  2. Intensity of radiation peaks at a specific wavelength that shifts with temperature (Wien’s Law).

Page 15: Classical Attempts to Explain Blackbody Radiation

1. Wien's Law: Explained spectral distribution for lower wavelengths but failed for higher wavelengths.

Page 16: Rayleigh-Jeans Law for Blackbody Radiation

• Rayleigh-Jeans Law: Describes energy density of blackbody radiation, agrees for long wavelengths but leads to ultraviolet catastrophe at short wavelengths.

Page 17: Ultraviolet Catastrophe

• Contradiction: Classical physics predicted infinite energy output at short wavelengths, an inconsistently termed 'ultraviolet catastrophe.'

Page 18: Max Planck and Quantum Theory

• Contribution: Planck introduced quantization of energy to explain blackbody radiation; discovered quantized energy exchange (hν).

Page 19: Planck's Radiation Law

• Quantization Insight: Confirmed emission and absorption of radiation must occur in discrete packets of energy (quanta).

Page 20: Quantum of Energy and Photons

• Photon Definition: Discrete packets of energy relating to electromagnetic radiation, absorption, and emission processes.

Page 21: Planck's Radiation Formula

• Expression: Formula accurately describes spectrums observed experimentally, encapsulating the nature of blackbody radiation using quantum concepts.

Page 22: Photoelectric Effect Overview

• Nobel Prize for Einstein (1921): For explaining the photoelectric effect, linking light and matter's particle nature.

Page 23: Photoelectric Effect Description

• Definition: Emission of electrons (photoelectrons) when light of sufficient frequency strikes a metal surface.

• Current Generation: Resulting flow of electrons forms photocurrent, observed by Heinrich Hertz in 1887.

Page 24: Laws of Photoelectric Emission

1. Threshold Frequency: Minimum frequency needed to release photoelectrons.

2. Instantaneous Emission: Emission begins immediately upon light exposure.

3. Kinetic Energy Relation: Max kinetic energy of emitted electrons is directly proportional to light frequency.

4. Current Intensity Relation: Strength of photocurrent relates to intensity, independent of frequency.

Page 25: Failures of Classical Theory in Explaining Photoelectric Effect

• Classical Limitations:

  • Failures to account for instantaneous emission and threshold frequency.

  • Cannot reconcile intensity and frequency effects on kinetic energy of emitted electrons.

Page 26: Quantum Analysis of Photoelectric Effect

• Einstein's Extension: Light consists of photons carrying quantized energy (hv) impacting metal's electrons.

• Photoelectric Equation: Wo = hvo (work function) relates minimum energy to liberate electrons and accounts for kinetic energy.

Page 27: Implications of Quantum Theory on Photoelectric Effect

• Electrons Emitted: Proportional to the number of photons striking the metal surface.

• Instantaneous Effect: Emission occurs quickly, reinforcing the quantum explanation of the phenomenon.

Page 28: Compton Effect Introduction

• Arthur Compton: American physicist awarded the Nobel Prize in 1927 for discovering the Compton effect.

Page 29: Illustration of Compton Effect

• Definition: Change in wavelength of X-rays after colliding with electrons, confirming particle-like behavior of light.

• Compton Shift Formula: λ' - λ = h/mc(1 - cos θ) - wavelength change as a function of scattering angle.

Page 30: Compton Effect Dynamics

• Pre and Post Collision: Describes energy and momentum before and after interactions.

• Longer Scattered Wavelengths: Demonstrates light behaving as particles, confirming duality.

Page 31: Compton Shift Overview

• Max Scattering Angle Cases: Highlights maximum possible shift in photon wavelengths during collisions, demonstrating wave-like properties.

Page 32: Significance of Compton Effect

• Milestone Achievement: Confirms electromagnetic radiation's particle nature and supports quantization of energy and momentum.

Page 33: Types of Waves Overview

• Matter Waves: Associated with particles possessing mass.

• Electromagnetic Waves: Oscillating electric and magnetic fields.

• Mechanical Waves: Oscillation of matter.

Page 34: De Broglie's Hypothesis

• Wave Nature of Matter: Proposed that all matter exhibits wave properties, validated by subsequent experiments.

Page 35: Proof of Wave-Particle Duality

• Photon Energy Relation: Establishes connections via Planck's constant (E = hv) for energy and momentum equations.

Page 36: Duality Equation Derivation

• Matter-Wave Relationship: Unifying equations relating momentum and wavelength with particle characteristics.

Page 37: De Broglie Wavelength Generalization

• General De Broglie Relation: λ = h/p, allowing description of all matter and wave characteristics.

Page 38: Application for Electrons

• Electron Wave Equations: Specific calculations for electrons subjected to varying potential differences, establishing connections with kinetic energy.

Page 39: De Broglie Wavelength Equation

• Calculation of Wavelength: Established relationship between de Broglie wavelength and potential energy, providing quantitative insights.

Page 40: Heisenberg’s Uncertainty Principle

• Foundation: Developed by Werner Heisenberg, illustrating inherent uncertainties in position and momentum measurements in quantum mechanics.

Page 41: Understanding Uncertainty

• Mathematical Representation: Highlighting the limitations of simultaneous measurement of position and momentum of quantum particles.

Page 42: Statement of Uncertainty Principle

• Precision Limitations in Measurement: Precise definition and mathematical formulation explaining fundamental constraints in particle physics.

Page 43: Applications of Uncertainty Principle

• Relation to Energy and Time: Expressed equivalently for uncertainties in energy levels and their correlations with time.

Page 44: Wave Functions in Quantum Mechanics

• Function Definition: Represents the probability amplitude describing the behavior of quantum particles in space and time dimensions.

Page 45: Characteristics of Well-Behaved Functions

• Normalization and Continuity: Conditions that must be met for the wave functions utilized in quantum mechanics calculations.

Page 46: Probability Density in Quantum Mechanics

• Definition: Represents the likelihood of locating a particle within a certain volume as derived from wave functions.

Page 47: Normalization of Wave Functions

• Integral Conditions: Establishing the necessary conditions for wave functions to ensure total probability equals unity.

Page 48: Operators in Quantum Mechanics

• Operator Definition: Mathematical constructs that manage quantum variables, yielding corresponding physical quantities upon application.

Page 49: Eigen Functions and Eigen Values

• Eigen Concept Explanation: Functions that yield consistent scaler values upon application of operators in quantum mechanics.

Page 50: Momentum Operator Description

• Mathematical Representation: Defined as acting upon wave functions to reveal momentum values and relationships concerning positional vectors.

Page 51: Energy Operator Explanation

• Energy Operator Definition: Illustrates the correlations between energy measurements in wave functions and quantized state relationships.

Page 52: Schrödinger’s Wave Equation Overview

• Schrödinger's Contribution: Describes the time-dependent behaviors of wave functions in quantum physics.

Page 53: Two Schrödinger Equations

• Types: 1. Time Dependent, 2. Time Independent; each addressing different system dynamics in quantum mechanics.

Page 54: Eigen Values of Schrödinger’s Wave Equation

• Physical Significance: Clarifies how eigen values correspond to allowable energy states in a quantum system.

Page 55: Time-Dependent Schrödinger’s Wave Equations

• Case of Free Particles: Math defining particle movement through oscillatory functions and the time evolution of momentum and energy properties.

Page 56: Time Dependent Schrödinger’s Solutions

• Equations and Momentum Linkage: Connection elucidated between kinetic energy, potential energy, and the fundamental wave equations.

Page 57: Formulation of Schrödinger's Equation

• Dynamic Compilation: Presenting combined energy, potential energies to derive the time-dependent equation describing behavior within force fields.

Page 58: Time Independent Case Description

• Independent Potential Energy Application: Helps simplify discussions around particles when external influences can be considered static, yielding clearer formulations.

Page 59: Full Time-Independent Schrödinger Equation

• General Representation in 3D: Formulation describing particles influenced by potentials solely dependent on spatial coordinates.

Page 60: 1D Infinite Potential Well Model

• Conceptual Foundation: Analyzing particles confined within quantum wells, illustrating discrete states and asymptotic freedoms within defined boundaries.

Page 61: Solving the Time-Independent Equation

• Separation of Variables Method: Use of distinct time and position-based functions to analyze energy stability under fixed potentials.

Page 62: Eigenvalue and Eigenfunction Relationships

• Final Representation of Energy Levels: Connections drawn between operators, observable energy states, and functions correlating with discrete levels within quantum contexts.

Page 63: 3D Quantum Box Solutions

• General Eigenvalue Relation: Affirmation of three-dimensional spatial dependencies for particle behavior under fixed border conditions.

Page 64: Dynamics in 1D Infinite Well

• Mathematical Equation Dynamics: Setup of conditions for particle oscillations in confined dimensions, yielding frequency correlational formulas.

Page 65: Finding Wave Function Values

• Boundary Conditions Established: Education on normalization process leading to the determination of wave functions in confined problems.

Page 66: Energy Level Quantization and Functions

• Energy Discreteness: Differentiating permitted energy levels compared to classical expectations, reinforcing the quantized views promoted by quantum mechanics.

Page 67: Node Variation in Wave Functions

• Visualization of Wave Properties: Understanding how wave functions relate to nodes—crucial for pinpointing quantized particle states within defined limits.

Page 68: Ensuring Wave Function Validity

• Normalization Importance: Enforcing procedures on wave equations ensures physical relevancy of computed results in quantum settings.

Page 69: Summary of Discrete Energy Levels

• Significance of Energy States: Clarifying quantum implications for energy exchange leading to node generation and boundary condition definitions.

Page 70: Visualization of Energy Levels in Quantum Mechanics

• Graphical Representation: Spatial modeling showcasing quantum levels and wave function distributions in particle assessments.

Page 71: Introduction to Quantum Tunneling

• Quantum Phenomenon: Detailed elucidation defining tunneling concepts that allow particles to breach barriers despite classical restrictions.

Page 72: Tunneling Dynamics Explained

• Energy Barriers: Analysis on thickness, mass, and impact of potential barriers affecting particle tunneling probabilities.

Page 73: Role of Quantum Tunneling in Fusion

• Star Energy Production: Framework detailing how quantum tunneling mediates nuclear fusion processes, fueling star lifecycles.

Page 74: α-Decay Mechanism through Tunneling

• Quantum Tunneling Application: Evaluation of how quantum mechanics satisfactorily explains α-decay processes within atomic structures.

Page 75: Quantum Tunneling Applications

• Technological Examples: Overview of technology such as scanning tunneling microscopy that harnesses quantum tunneling principles for practical applications.

Page 76: Understanding Finite Potential Barriers

• Barrier Definitions: Evaluation of particle behavior in relation to well-defined finite potentials and their geometric parameters, critical for tunneling phenomena.

Page 77: Transmission Probability Equations

• Quantifying Tunneling: Mathematical representation illustrating factors affecting particle penetration probabilities correlating to barrier properties.