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The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector , the state vector . If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
The Heisenberg picture is closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly correspond to classical Poisson brackets). The Schrödinger picture, the preferred formulation in introductory texts, is easy to visualize in terms of Hilbert space rotations of state vectors, although it ...
At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state.
In quantum mechanics a state space is a separable complex Hilbert space.The dimension of this Hilbert space depends on the system we choose to describe. [1] [2] The different states that could come out of any particular measurement form an orthonormal basis, so any state vector in the state space can be written as a linear combination of these basis vectors.
Since | is a constant ket (the state ket at t = 0), and since the above equation is true for any constant ket in the Hilbert space, the time evolution operator must obey the equation = (). If the Hamiltonian is independent of time, the solution to the above equation is [ note 1 ] U ( t ) = e − i H t / ℏ . {\displaystyle U(t)=e^{-iHt/\hbar }.}
Indeed, the state of the system in the atomic and subatomic scale is described not by dynamic variables with specific numerical values, but by state functions that are dependent on the c-number time. In this realm of quantum systems, the equation of motion governing dynamics heavily relies on the Hamiltonian, also known as the total energy ...
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the phase space usually consists of all possible values of the position and momentum parameters.
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states , both mixed states and pure states .