Probability amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems.These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein.It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.The system may always be described by a linear combination or superposition of these eigenstates with unequal "weights".In other words, the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements.A second, subsequent observation of Q no longer certainly produces the eigenvalue corresponding to the starting state.In a formal setup, the state of an isolated physical system in quantum mechanics is represented, at a fixed time, by a state vector |Ψ⟩ belonging to a separable complex Hilbert space.of the Hilbert space can be written as[1] Its relation with an observable can be elucidated by generalizing the quantum stateAs probability is a dimensionless quantity, |ψ(x)|2 must have the inverse dimension of the variable of integration x.Whereas a Hilbert space is separable if and only if it admits a countable orthonormal basis, the range of a continuous random variableAs such, eigenstates of an observable need not necessarily be measurable functions belonging to L2(X) (see normalization condition below).The amplitudes are composed of state vector |Ψ⟩ indexed by A; its components are denoted by ψ(x) for uniformity with the previous case.A convenient configuration space X is such that each point x produces some unique value of the observable Q.[citation needed] Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator.[clarification needed] An example of the discrete case is a quantum system that can be in two possible states, e.g. the polarization of a photon.), the following must be true for the measurement of spin "up" and "down": If one assumes that system is prepared, so that +1 is registered inThis leads to a constraint that α2 + β2 = 1; more generally the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one.One can always divide any non-zero element of a Hilbert space by its norm and obtain a normalized state vector.Suppose a wave function ψ(x, t) gives a description of the particle (position x at a given time t).A wave function is square integrable if After normalization the wave function still represents the same state and is therefore equal by definition to[5][6] Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle.[7] This is key to understanding the importance of this interpretation: for a given particle constant mass, initial ψ(x, t0) and potential, the Schrödinger equation fully determines subsequent wavefunctions.When no measurement apparatus that determines through which slit the electrons travel is installed, the observed probability distribution on the screen reflects the interference pattern that is common with light waves.A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account.However, one may choose to devise an experiment in which the experimenter observes which slit each electron goes through.One may go further in devising an experiment in which the experimenter gets rid of this "which-path information" by a "quantum eraser".[8] Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.The concept of amplitudes is also used in the context of scattering theory, notably in the form of S-matrices.Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of matrix elements squared are interpreted as transition probabilities just as in a random process.Like a finite-dimensional unit vector specifies a finite probability distribution, a finite-dimensional unitary matrix specifies transition probabilities between a finite number of states.
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