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    Quantum Field Theory

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    • Introduction to Quantum Mechanics
      • 1.1Historical Background
      • 1.2Introduction to Quantum Concepts
      • 1.3Quantum States and Observables
    • Wave-Particle Duality
      • 2.1The Double Slit Experiment
      • 2.2Heisenberg's Uncertainty Principle
      • 2.3Quantum Superposition and Entanglement
    • The Schrödinger Equation
      • 3.1Time-Dependent Equation
      • 3.2Stationary States
      • 3.3Square Well Potential
    • Quantum Operators and Measurement
      • 4.1Quantum Operators
      • 4.2The Measurement Postulate
      • 4.3Complex Probability Amplitudes
    • Quantum Mechanics of Systems
      • 5.1Quantum Harmonic Oscillator
      • 5.2Quantum Angular Momentum
      • 5.3Particle in a Box
    • The Dirac Equation
      • 6.1Wave Equations
      • 6.2The Dirac Sea
      • 6.3Hole Theory
    • Introduction to Quantum Electrodynamics (QED)
      • 7.1Electromagnetic Field
      • 7.2Feynman Diagrams
      • 7.3QED Interactions
    • Path Integrals and Quantum Mechanics
      • 8.1Feynman’s Approach
      • 8.2Action Principle
      • 8.3Quantum Oscillator Problem
    • Symmetries in Quantum Field Theory
      • 9.1Gauge Symmetry
      • 9.2Poincaré Symmetry
      • 9.3Global and Local Symmetries
    • Quantum Chromodynamics
      • 10.1Color Charge
      • 10.2Quark Model
      • 10.3Confinement and Asymptotic Freedom
    • The Higgs Mechanism
      • 11.1Electroweak Symmetry Breaking
      • 11.2The Higgs Boson
      • 11.3Implication for Mass of Known Particles
    • Quantum Field Theory in Curved Space-Time
      • 12.1The Concept of Spacetime
      • 12.2Quantum Effects in Curved Spaces
      • 12.3Hawking Radiation
    • Quantum Cosmology and Conclusion
      • 13.1Big Bang Theory
      • 13.2Cosmic Inflation
      • 13.3Looking Ahead: Frontiers in Quantum Mechanics

    Path Integrals and Quantum Mechanics

    Understanding the Action Principle in Quantum Mechanics

    formal sum or integral over all histories of a quantum system

    Formal sum or integral over all histories of a quantum system.

    The action principle, also known as the principle of least action, is a cornerstone of theoretical physics. It is a principle that states that the path taken by a physical system is such that the action is minimized. This principle is a key component in the path integral formulation of quantum mechanics, which was developed by Richard Feynman.

    Introduction to the Principle of Least Action

    The principle of least action is a variational principle. According to this principle, the path that a physical system will take between two states is the one for which the action is minimized. The action is a quantity that is calculated by integrating the Lagrangian over time.

    The Role of the Action Principle in the Path Integral Formulation

    In the path integral formulation of quantum mechanics, the action principle plays a crucial role. The path integral formulation sums over all possible histories of a system, and each history contributes to the sum with a weight that is determined by the action. The histories with the least action contribute the most to the sum.

    The Lagrangian and Hamiltonian in the Action Principle

    The Lagrangian and Hamiltonian are two quantities that are central to the action principle. The Lagrangian is a function that describes the dynamics of a system. It is defined as the difference between the kinetic and potential energy of the system. The action is calculated by integrating the Lagrangian over time.

    The Hamiltonian, on the other hand, is the total energy of the system. It is defined as the sum of the kinetic and potential energy. The Hamiltonian plays a crucial role in the Schrödinger equation, which is the fundamental equation of quantum mechanics.

    The Euler-Lagrange Equation and Its Role in the Action Principle

    The Euler-Lagrange equation is a differential equation that is derived from the action principle. It provides the equations of motion for a system. The Euler-Lagrange equation is obtained by setting the variation of the action to zero. This equation is used to find the path that minimizes the action, which is the path that the system will take according to the action principle.

    In conclusion, the action principle is a fundamental concept in theoretical physics and quantum mechanics. It provides a powerful tool for understanding the dynamics of physical systems.

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