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Semester 3: Core Paper-8 Quantum Mechanics II
Scattering Theory - Scattering amplitude, Born approximation, optical theorem
Scattering Theory
Scattering Amplitude
Scattering amplitude represents the probability amplitude for a scattering process. It is a complex quantity related to the differential cross-section of an interaction. The amplitude provides essential insights into the interaction potential and the behavior of particles during collisions.
Born Approximation
The Born approximation is a method used to simplify the calculations in scattering problems, particularly when the potential is weak. It assumes that the incident wave function remains unchanged and that the first-order term in perturbation theory accurately describes the scattering process. This is particularly effective for high-energy collisions where the potential has a negligible effect on the scattering.
Optical Theorem
The optical theorem relates the total cross-section of a scattering process to the imaginary part of the forward scattering amplitude. It provides a crucial connection between scattering theory and experimental measurements. This theorem is significant because it allows physicists to infer total scattering probabilities from measurements made in the forward direction.
Perturbation Theory - Time dependent perturbations, Fermi golden rule, selection rules
Perturbation Theory in Quantum Mechanics
Introduction to Perturbation Theory
Perturbation theory is a mathematical technique used to find an approximate solution to a problem that cannot be solved exactly. In quantum mechanics, it is applied when a Hamiltonian can be divided into two parts: a solvable part and a small perturbing part.
Time-Dependent Perturbation Theory
Time-dependent perturbation theory deals with systems where the Hamiltonian changes with time. It is used to study transitions between states due to time-dependent interactions, allowing the calculation of transition probabilities and expectation values over time.
Fermi's Golden Rule
Fermi's golden rule gives the probability of a transition between states per unit time due to a perturbation. It is derived from time-dependent perturbation theory and is widely used in quantum mechanics to predict rates of transitions between quantum states.
Selection Rules
Selection rules are guidelines that determine the allowed transitions between quantum states. They arise due to symmetries in the system and the nature of the perturbation. These rules help predict whether a transition can occur and which transitions are favored based on conservation laws.
Relativistic Quantum Mechanics - Klein-Gordon and Dirac equations, negative energy states, antiparticles
Relativistic Quantum Mechanics
Klein-Gordon Equation
The Klein-Gordon equation is a relativistic wave equation derived from combining quantum mechanics and special relativity. It describes scalar particles and is the simplest equation that incorporates relativistic effects. The equation is expressed as: (□ + m²)ψ = 0, where □ is the d'Alembert operator and m is the mass of the particle.
Dirac Equation
The Dirac equation provides a description of fermionic particles and incorporates both quantum mechanics and special relativity. It predicts the existence of spin-1/2 particles and provides a framework for understanding electron behavior. The equation is expressed as: (iγ^μ∂_μ - m)ψ = 0, where γ^μ are the gamma matrices.
Negative Energy States
In the context of the Klein-Gordon equation, negative energy solutions arise, leading to difficulties in interpreting physical states. These solutions imply the existence of states with negative energy, which could lead to instabilities in a physical system.
Antiparticles
The Dirac equation introduced the concept of antiparticles, which are counterparts to particles that possess the same mass but opposite charge. For example, the positron is the antiparticle of the electron. The existence of antiparticles is a significant outcome of relativistic quantum mechanics and has been experimentally confirmed.
Dirac Equation - Gamma matrices, relativistic invariance, probability current
Dirac Equation
Overview of the Dirac Equation
The Dirac Equation is a fundamental equation in quantum mechanics that describes fermions, such as electrons. It was formulated by Paul Dirac in 1928 and is a key component of quantum field theory. The equation incorporates both quantum mechanics and special relativity, providing a relativistic description of particles with spin-1/2.
Gamma Matrices
Gamma matrices are a set of matrices that are used in the formulation of the Dirac Equation. They satisfy the Clifford algebra and are essential in expressing the equation in a compact form. Each gamma matrix corresponds to a specific direction in spacetime and allows for the inclusion of spin and relativistic effects. The matrices are denoted as gamma(0), gamma(1), gamma(2), and gamma(3) in four-dimensional spacetime.
Relativistic Invariance
The Dirac Equation ensures that the equations of motion for particles remain invariant under Lorentz transformations, which are fundamental to the theory of relativity. This property signifies that the physics described by the equation does not change for observers in different inertial frames. This invariance is crucial for ensuring the consistency of quantum mechanics with special relativity.
Probability Current
In the context of the Dirac Equation, the concept of probability current is introduced to maintain the probabilistic interpretation of quantum mechanics. The probability current density is derived from the wave function and is used to describe the flow of probability in space and time. It is essential for ensuring conservation of probability, similar to how probability density evolves according to the continuity equation.
Classical Fields and Second Quantization - Euler-Lagrange equation, quantization of fields, creation and annihilation operators
Classical Fields and Second Quantization
Euler-Lagrange Equation
The Euler-Lagrange equation is a fundamental equation in the calculus of variations. It is used to derive the equations of motion for a system described by a Lagrangian. The Lagrangian is a function that summarizes the dynamics of the system, often expressed in terms of the kinetic and potential energies. The Euler-Lagrange equation is given by the formula: d/dt(∂L/∂(dq/dt)) - ∂L/∂q = 0, where L is the Lagrangian, q is the generalized coordinate, and dq/dt is the generalized velocity. This equation provides a way to obtain the equations of motion from the Lagrangian framework.
Quantization of Fields
Quantization of fields is the process of promoting classical fields to quantum operators, a key step in the transition from classical to quantum field theory. The process involves expressing the classical field in terms of creation and annihilation operators, which correspond to the addition and removal of quantum excitations (particles). The quantization condition requires that fields satisfy certain commutation relations to reflect the underlying quantum mechanics. This leads to the formulation of quantum field theories that describe particle interactions at a fundamental level.
Creation and Annihilation Operators
Creation and annihilation operators are crucial components in quantum field theory. The creation operator, typically denoted as a†, adds a quantum of excitation to a given state, effectively increasing the number of particles in that state. Conversely, the annihilation operator, represented as a, removes a quantum of excitation, decreasing the particle count. In terms of a quantum harmonic oscillator, these operators facilitate the construction of the Fock space, which provides a mathematical representation of states with variable particle numbers. These operators obey specific commutation relations that are fundamental to the underlying statistics of the particles, such as bosons and fermions.
