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
Failures of Classical Mechanics:
Stability/Structure of Atom
Blackbody Radiation
Photoelectric Effect
Compton Scattering
Wave-Particle Duality
Heisenberg's Uncertainty Principle
Wave Function and Its Physical Significance
Operators and Eigen Functions
Energy and Momentum Operators
Schrödinger's Wave Equations
Time Dependent
Time Independent
Particle in a 1-D Infinite Well
Quantum Tunneling
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:
Stability and structure of atoms.
Blackbody radiation results.
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:
Non-uniform distribution across the spectrum at constant temperature.
Intensity of radiation peaks at a specific wavelength that shifts with temperature (Wien’s Law).
Page 15: Classical Attempts to Explain Blackbody Radiation
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
Threshold Frequency: Minimum frequency needed to release photoelectrons.
Instantaneous Emission: Emission begins immediately upon light exposure.
Kinetic Energy Relation: Max kinetic energy of emitted electrons is directly proportional to light frequency.
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.