Path Integral Formulation
Introduction
The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation was first developed by physicist Richard P. Feynman.
Historical Background
The path integral formulation is a cornerstone in the study of quantum field theory and statistical mechanics, both of which involve the computation of quantities under the action of forces. The formulation has found use in the study of solid state physics, quantum chemistry, nuclear physics, and particle physics. It is also a fundamental aspect of string theory and the theory of quantum gravity.
Formalism
The path integral formulation of quantum mechanics represents a pivot from the traditional formulation. It is a more general understanding of quantum mechanics, and it provides a method to compute quantum mechanical amplitudes directly without going through the process of finding wave functions and then squaring them to find probabilities.
Action Principle
The action principle is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. The action, usually denoted by S, is defined as the integral of the Lagrangian function (L) over time.
Quantum Amplitude
In the path integral formulation, the quantum amplitude for a particle to go from one place to another in a certain time is given by a path integral over all possible paths of the particle in its configuration space. The quantum amplitude is a complex number whose absolute square gives the probability.
Functional Integral
The functional integral, also known as the path integral, is the integral over all possible paths. Each path is considered to be a function, and the integral sums over all these functions. The integral is taken over a function space, which is an infinite-dimensional space in which each point is a function.
Applications
The path integral formulation has been used to calculate quantities in quantum mechanics, quantum field theory, and statistical mechanics. It has also been used to derive the Schrödinger equation, the equations of motion for quantum fields, and the statistical properties of quantum systems.
Quantum Field Theory
In quantum field theory, the path integral formulation extends to fields as well as particles. The fields are treated as a system of an infinite number of degrees of freedom, and the action is taken to be an integral over all space of a Lagrangian density function.
Statistical Mechanics
In statistical mechanics, the path integral formulation provides a way to calculate partition functions, which are used to compute thermodynamic quantities. The paths in the path integral represent possible states of the system, and the action is related to the energy of the system.
Quantum Gravity and String Theory
In theories of quantum gravity and string theory, the path integral formulation is used to calculate the probability amplitudes for different configurations of the gravitational field or strings.