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This is the complete lecture note for PHY 402 Quantum Mechanics II. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
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LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS COURSE CODE: PHY 402 COURSE TITLE: Quantum Mechanics II COURSE UNIT: 3 UNITS Here are five comprehensive series titles that fully cover the scope of the attached PHY 402 course: Series 1: Perturbation Methods in Quantum Mechanics Series 2: Quantum Scattering Theory and Elastic Potential Scattering Series 3: Symmetries and Selection Rules in Quantum Mechanics Series 4: Quantum Models of Atomic, Molecular, Solid-State, and Nuclear Phenomena Series 5: Relativistic Wave Equations and Quantum Motion in Electromagnetic Fields Series 2: Scattering Theory in Quantum Mechanics Part 2.1: Fundamentals of Scattering 2.1.1 Classical versus quantum scattering 2.1.2 Scattering amplitude and differential cross-section 2.1.3 Physical interpretation of scattering experiments Part 2.2: Elastic Potential Scattering 2.2.1 Concept of elastic scattering 2.2.2 Potential scattering models 2.2.3 Applications in atomic and nuclear systems Part 2.3: Partial Wave Analysis 2.3.1 Expansion in spherical harmonics 2.3.2 Phase shifts and resonance phenomena 2.3.3 Applications in low-energy scattering Part 2.4: Born Approximation and Green’s Function Methods 2.4.1 First Born approximation 2.4.2 Higher-order corrections 2.4.3 Green’s function approach to scattering problems Introduction In quantum mechanics, scattering experiments provide one of the most direct ways of probing the structure and interactions of particles. By studying how particles deflect when they encounter a potential, physicists gain valuable information about the forces at play and the nature of matter at microscopic scales. Scattering theory therefore forms a cornerstone of modern physics, linking theoretical models with experimental observations. The aim of this Series is to introduce you to the principles and methods of quantum scattering, with particular emphasis on elastic potential scattering. You will learn how scattering amplitudes and cross-sections are defined, how partial wave analysis is applied, and how approximation methods such as the Born approximation and Green’s function techniques are used to solve scattering problems. We shall commence this Series by examining the fundamentals of scattering, comparing classical and quantum approaches and introducing the concepts of scattering amplitude and cross-section.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Next, we will explore elastic potential scattering, focusing on how potential models describe collisions and their applications in atomic and nuclear systems. Following that, we will move on to partial wave analysis, where we will study spherical harmonic expansions, phase shifts, and resonance phenomena. Finally, we will conclude with the Born approximation and Green’s function methods, which provide powerful tools for solving complex scattering problems and connecting theory with experimental data. Series Learning Outcomes Series 2: Scattering Theory in Quantum Mechanics Upon completing this Series, you should be able to: SLO 2.1 define the basic concepts of scattering in quantum mechanics and distinguish them from classical scattering, SLO 2.2 explain the meaning of scattering amplitude and differential cross-section and interpret their physical significance, SLO 2.3 describe the principles of elastic potential scattering and apply potential models to atomic and nuclear systems, SLO 2.4 demonstrate the use of partial wave analysis by expanding scattering states in spherical harmonics and identifying phase shifts, SLO 2.5 analyse resonance phenomena in low-energy scattering using partial wave methods, SLO 2.6 apply the Born approximation to calculate scattering amplitudes for weak potentials, and SLO 2.7 evaluate the role of Green’s function methods in solving scattering problems and connecting theory with experimental data. 2.1 Fundamentals of Scattering Scattering is one of the most important tools in quantum mechanics because it allows us to probe the structure and interactions of particles. By studying how particles deflect when they encounter a potential, we gain insight into the forces acting at microscopic scales. Unlike classical scattering, quantum scattering requires us to consider the wave nature of particles, leading to concepts such as scattering amplitudes and cross-sections. 2.1.1 Classical versus quantum scattering In classical scattering, particles are treated as point-like objects that follow definite trajectories when they interact with a potential. The deflection angle depends on the impact parameter and the nature of the force. In quantum scattering, particles are described by wave functions. Instead of definite trajectories, we calculate probabilities for particles to scatter into different directions. Classical scattering: deterministic paths. Quantum scattering: probabilistic outcomes. Fill in the blank: In quantum scattering, particles are described by ______ functions rather than definite trajectories.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Figure 2.1 Caption: Comparison of classical and quantum scattering. 2.1.2 Scattering amplitude and differential cross-section The scattering amplitude is a complex quantity that encodes the probability amplitude for a particle to scatter into a particular direction. The differential cross-section is defined as: dσdΩ=∣f(θ,ϕ)∣2 where f(θ,ϕ) is the scattering amplitude. Physically, the differential cross-section tells us how likely a particle is to scatter into a small solid angle dΩ. Fill in the blank: The differential cross-section is the square of the ______ amplitude. Box 2.1 Placeholder: Key features of scattering amplitude and cross-section. 2.1.3 Physical interpretation of scattering experiments Scattering experiments are used to study the structure of matter.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS By measuring cross-sections, physicists infer the nature of forces and potentials. Elastic scattering preserves the kinetic energy of the particles, while inelastic scattering involves energy transfer. Examples include Rutherford’s experiment, which revealed the nuclear structure of atoms. Fill in the blank: Rutherford’s scattering experiment revealed the existence of the atomic ______. Figure 2.2 Caption: Rutherford’s experiment and its role in discovering the atomic nucleus. Pilot questions 2.1 What is the main difference between classical and quantum scattering? Define scattering amplitude. Write down the formula for the differential cross-section. What does the differential cross-section physically represent? Which famous experiment revealed the existence of the atomic nucleus? 2.2 Elastic Potential Scattering Elastic potential scattering is a central topic in quantum scattering theory. In this process, particles interact with a potential but emerge with the same total kinetic energy they had before the collision. Unlike inelastic scattering, no energy is transferred to internal degrees of freedom, so the scattering is purely a redirection of motion. 2.2.1 Concept of elastic scattering Elastic scattering occurs when the incident particle collides with a target and is deflected without any loss of kinetic energy. The wave function of the scattered particle retains the same energy as the incoming wave. Only the direction of motion changes, not the magnitude of the momentum. Elastic scattering is often the simplest case to analyse and forms the basis for more complex scattering processes. Fill in the blank: In elastic scattering, the total kinetic energy of the particle is ______ after the collision.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Figure 2.3 Caption: Elastic scattering preserves kinetic energy while changing direction. 2.2.2 Potential scattering models In quantum mechanics, elastic scattering is often studied using potential scattering models, where the interaction is represented by a potential function V(r). Common potentials include square wells, Coulomb potentials, and Yukawa potentials. The Schrödinger equation is solved with the potential to obtain scattering amplitudes and cross-sections. These models help us understand how different forces influence scattering behaviour. Fill in the blank: In potential scattering models, the interaction is represented by a ______ function. Box 2.2 Placeholder: Examples of potential scattering models. 2.2.3 Applications in atomic and nuclear systems Elastic potential scattering has wide applications in physics: Atomic systems: Electron scattering from atoms reveals information about atomic orbitals and charge distributions.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Nuclear systems: Neutron scattering from nuclei provides insights into nuclear structure and forces. Condensed matter: Elastic scattering of electrons in crystals helps determine lattice structures. Fill in the blank: Elastic scattering of neutrons from nuclei provides insights into ______ structure. Figure 2.4 Caption: Neutron scattering reveals nuclear structure through elastic scattering. Pilot questions 2.2 What is elastic scattering, and how does it differ from inelastic scattering? In elastic scattering, what happens to the total kinetic energy of the particle? What are potential scattering models, and why are they useful? Give two examples of potentials used in scattering models. How is elastic scattering applied in atomic and nuclear systems? 2.3 Partial Wave Analysis Partial wave analysis is a powerful method in quantum scattering theory. It allows us to express the scattering wave function as a sum of spherical harmonics, each corresponding to a different angular momentum component. This approach is particularly useful for central potentials, where the interaction depends only on the distance between particles. 2.3.1 Expansion in spherical harmonics In quantum mechanics, the scattered wave function can be expanded in terms of spherical harmonics, which are functions that describe angular dependence. The incident plane wave is expressed as a sum of spherical waves. Each spherical wave corresponds to a particular angular momentum quantum number l. This expansion simplifies the analysis of scattering by reducing the problem to radial equations. Fill in the blank: In partial wave analysis, the angular dependence of the wave function is expressed using ______ harmonics.
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