Quantum state

In quantum physics, a quantum state is the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, i.e. for the outcome of each possible measurement on the system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. In quantum physics, a quantum state is the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, i.e. for the outcome of each possible measurement on the system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, all other states are called mixed quantum states. Mathematically, a pure quantum state can be represented by a ray in a Hilbert space over the complex numbers. The ray is a set of nonzero vectors differing by just a complex scalar factor; any of them can be chosen as a state vector to represent the ray and thus the state. A unit vector is usually picked, but its phase factor can be chosen freely anyway. Nevertheless, such factors are important when state vectors are added together to form a superposition. Hilbert space is a generalization of the ordinary Euclidean space:93–96 and it contains all possible pure quantum states of the given system. If this Hilbert space, by choice of representation (essentially a choice of basis corresponding to a complete set of observables), is exhibited as a function space (a Hilbert space in its own right), then the representatives are called wave functions. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin z-component sz. A more complicated case is given (in bra–ket notation) by the singlet state: | ψ ⟩ = 1 2 ( | ↑↓ ⟩ − | ↓↑ ⟩ ) , {displaystyle left|psi ight angle ={frac {1}{sqrt {2}}}{igg (}left|uparrow downarrow ight angle -left|downarrow uparrow ight angle {igg )},} which involves superposition of joint spin states for two particles with spin ​1⁄2. As suggested by the previous equation, the singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability. Remarkably, this holds even if the measurements are made at the same time and the particles are arbitrarily far apart. A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. Mixed states are described by so-called density matrices. A pure state can also be recast as a density matrix; in this way, pure states can be represented as a subset of the more general mixed states. For example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional. A pure state here is represented by a two-dimensional complex vector ( α , β ) {displaystyle (alpha ,eta )} , with a length of one; that is, with