Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales, such as atoms and subatomic particles. It introduces concepts like wave-particle duality, uncertainty principle, and quantum entanglement, which challenge classical intuitions about the behavior of matter and energy.
Lagrangian density is a function that encapsulates the dynamics of a field in classical and quantum field theories, providing the equations of motion through the Euler-Lagrange equations. It is the field-theoretic generalization of the Lagrangian function used in classical mechanics, and its integral over space yields the action, a central quantity in determining the evolution of physical systems.
The Feynman Path Integral is a formulation of quantum mechanics that expresses the probability amplitude for a particle's state as a sum over all possible paths the particle could take, weighted by an exponential of the classical action. This approach provides deep insights into quantum field theory and has applications in various areas of physics, including statistical mechanics and quantum gravity.
Functional differentiation is a mathematical process used to determine the derivative of a functional, which is a mapping from a space of functions to the real numbers. This concept is crucial in fields like calculus of variations and quantum mechanics, where it helps in finding functions that optimize certain criteria or describe physical systems.
Path Integral Quantization is a formulation of quantum mechanics that generalizes the action principle of classical mechanics, allowing for the calculation of quantum amplitudes by summing over all possible paths a particle can take. It provides a powerful framework for understanding quantum field theory and the transition from quantum to classical behavior through the principle of least action.