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Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems . A statistical ensemble is described by a density operator S , which is a non-negative, self-adjoint , trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics .
Quantum statistical mechanics
Quantum field theory
Quantum field theory ( QFT ) is a theoretical framework for constructing quantum mechanical models of fields and many-body systems (in a condensed matter context), both of which are systems classically represented by an infinite number of degrees of freedom . They are also used in the description of critical phenomena and quantum phase transitions , such as in the BCS theory of superconductivity . Quantum field theories are especially useful for describing systems where the particle count/number may change over the course of a reaction. Most theories in modern particle physics are formulated as relativistic quantum field theories, such as quantum electrodynamics (QED), quantum chromodynamics (QCD), and the Standard Model .
In quantum mechanics , the Schrödinger equation is a partial differential equation that describes how the quantum state of some physical system changes with time . It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger . In classical mechanics , the equation of motion is Newton's second law , and equivalent formulations are the Euler–Lagrange equations and Hamilton's equations . In all these formulations, they are used to solve for the motion of a mechanical system, and mathematically predict what the system will do at any time beyond the initial settings and configuration of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system, usually atoms, molecules, and subatomic particles; free, bound, or localized.
Schrödinger equation
Scattering theory
Top: the real part of a plane wave travelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk of index of refraction higher than the index of the surrounding medium. This object scatters part of the wave field, although at any individual point, the wave's frequency and wavelength remain intact. In mathematics and physics , scattering theory is a framework for studying and understanding the scattering of waves and particles .The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics . It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral , over an infinity of possible trajectories to compute a quantum amplitude . The basic idea of the path integral formulation can be traced back to Norbert Wiener , who introduced the Wiener integral for solving problems in diffusion and Brownian motion . [ 1 ] This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 paper. [ 2 ] The complete method was developed in 1948 by Richard Feynman .
Path integral formulation