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In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while ...
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.
A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a phase space formulated by symplectic manifold, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are ...
The postulates of quantum mechanics state that pure states, at a given time t, correspond to vectors in a separable complex Hilbert space, while each measurable physical quantity (such as the energy or momentum of a particle) is associated with a mathematical operator called the observable.
As the phase varies, the coherent state circles around the origin and the disk neither distorts nor spreads. This is the most similar a quantum state can be to a single point in phase space. Figure 5: Phase space plot of a coherent state. This shows that the uncertainty in a coherent state is equally distributed in all directions.
By Gelfand representation, every commutative C*-algebra A is of the form C 0 (X) for some locally compact Hausdorff X. In this case, S(A) consists of positive Radon measures on X, and the pure states are the evaluation functionals on X. More generally, the GNS construction shows that every state is, after choosing a suitable representation, a ...
For a two-dimensional Hilbert space, the space of all such states is the complex projective line. This is the Bloch sphere, which can be mapped to the Riemann sphere . The Bloch sphere is a unit 2-sphere , with antipodal points corresponding to a pair of mutually orthogonal state vectors.
The parity operator is defined by its action in the | representation of changing r to −r, i.e. | | = The eigenvalues of P can be shown to be limited to , which are both degenerate eigenvalues in an infinite-dimensional state space. An eigenvector of P with eigenvalue +1 is said to be even, while that with eigenvalue −1 is said to be odd.