Space in phases
Introduction
In classical mechanics, the phase space, phase space or phase diagram is a mathematical construction that allows representing the set of positions and for their respective moments.
The phase space formalism is used in the context of Lagrangian mechanics and Hamiltonian mechanics. Phase space or a part of it is usually designated by Γ (capital gamma). Physically each point of mechanical space.
In statistical physics, probability distributions defined over phase space are used. Starting from a certain subset of the probability distributions of a phase space, a Hilbert space structure can be constructed. These spaces are used in quantum mechanics.
Phasic space in classical mechanics
In classical mechanics, phase space is a mathematical construction based on configuration space. Specifically, an adequate phase space for a system with a finite number of degrees of freedom is the cotangent bundle of the configuration space of the mechanical system.
This cotangent bundle constructed in this way can also be provided with a symplectic topology where the theorems of Hamiltonian mechanics can be conveniently formulated.
One of the classical theorems on phase spaces is Liouville's theorem "Liouville's theorem (Hamiltonian mechanics)"), according to which a cloud of points distributed according to a probability density representing a macroscopic equilibrium state ρ(p,q) must be invariant in time.
Furthermore, each Hamiltonian H defined on a phase space is associated with a set of time evolution trajectories. The set of trajectories constitutes a one-dimensional foliation of the phase space that covers almost the entire phase space (specifically the entire phase space, except for a set of null measure), the latter equivalent to the fact that the space can be decomposed into trajectories that do not intersect.
Phase space in quantum mechanics
Contenido
Uno de los rasgos definitorios de la mecánica cuántica es que el estado físico de un sistema no determina el resultado de cualquier medida que pueda hacerse sobre él. En términos más crudos, el resultado de una medida sobre dos sistemas cuánticos que tienen el mismo estado físico no siempre arroja los mismos resultados.