For example, the values of the energy of a bound system are always discrete, and angular momentum components have values that take the form m ℏ, where m is either an integer or a half-integer, positive or negative. For the case of a system of particles, the space V consists of functions called wave functions or state vectors. The relation between the state of a quantum system and the value of an observable requires some linear algebra for its description. A crucial difference between classical quantities and quantum mechanical observables is that the latter may not be simultaneously measurable, a property referred to as complementarity. Since the eigenvalue of an observable represents a possible physical quantity that its corresponding dynamical variable can take, we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space. ⟩ Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic but statistically predictable way. Observables corresponding to non-commutative operators are called incompatible. the observable properties of a quantum system can be described in quantum mechanics, that is in terms of Hermitean operators. and {\displaystyle {\hat {A}}} {\displaystyle \mathbf {w} =c\mathbf {v} } The observables associated with such operators are said to be compatible. In quantum physics, observables manifest as linear operators on a Hilbert space representing the state space of quantum states. ⟩ ψ {\displaystyle {\hat {A}}} In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a Hilbert space V. Two vectors v and w are considered to specify the same state if and only if Observables can be represented by a Hermitian matrix if the Hilbert space is finite-dimensional. {\displaystyle |\psi _{a}\rangle } {\displaystyle \mathbf {A} } A This eigenket equation says that if a measurement of the observable stream C The eigenvalues of operator In classical physics, all quantities are compatible – we can measure any two quantities we like, and the measurements don't interfere with each other. that acts on the state of the quantum system. However, if the system of interest is in the general state In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. , with eigenvalue In an infinite-dimensional Hilbert space, the observable is represented by a symmetric operator, which may not be defined everywhere. a In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Then. ^ After one measured, for example λi the system will be in the state |ϕ′\rang=|ψi\rang, i.e. For example, suppose | Let O = O† be an observable with with discrete non-degenerate spectrum λ1, λ2, …, λn and has descrete eigenstates |ψi\rang i = 1, …, n. Now assume the system is prepared in a state |ϕ\rang, which can be represented in eigenbasis of the observable |ϕ\rang=∑ni=1ci|ψi\rang, where ci ∈ C Each measurement of the observable O will give some outcome λi with probability P(λi)=|ci|2=|\langψi|ϕ\rang|2. This is mathematically expressed by non-commutativity of the corresponding operators, to the effect that the commutator By the structure of quantum operations, this description is mathematically equivalent to that offered by relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.

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