Кванттық механикадағы ықтималдық амплитудасы

Complex number whose squared absolute value is a probability
probability amplitude in quantum mechanics

In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity represents a probability density. Probability amplitudes provide a relationship between the quantum state vector of a system and the results of observations of that system, a link was first proposed by Max Born, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding, and the probability thus calculated is sometimes called the "Born probability". 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.

Neglecting some technical complexities, the problem of quantum measurement is the behaviour of a quantum state, for which the value of the observable Q to be measured is uncertain. Such a state is thought to be a coherent superposition of the observable's eigenstates, states on which the value of the observable is uniquely defined, for different possible values of the observable. When a measurement of Q is made, the system (under the Copenhagen interpretation) jumps to one of the eigenstates, returning the eigenvalue belonging to that eigenstate. The system may always be described by a linear combination or superposition of these eigenstates with unequal "weights". Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the Born rule. Clearly, the sum of the probabilities, which equals the sum of the absolute squares of the probability amplitudes, must equal 1. This is the normalization requirement. If the system is known to be in some eigenstate of Q (e. g. after an observation of the corresponding eigenvalue of Q) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of Q (so long as no other important forces act between the measurements). In other words, the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. If the set of eigenstates to which the system can jump upon measurement of Q is the same as the set of eigenstates for measurement of R, then subsequent measurements of either Q or R always produce the same values with probability of 1, no matter the order in which they are applied. The probability amplitudes are unaffected by either measurement, and the observables are said to commute. By contrast, if the eigenstates of Q and R are different, then measurement of R produces a jump to a state that is not an eigenstate of Q. Therefore, if the system is known to be in some eigenstate of Q (all probability amplitudes zero except for one eigenstate), then when R is observed the probability amplitudes are changed. A second, subsequent observation of Q no longer certainly produces the eigenvalue corresponding to the starting state. In other words, the probability amplitudes for the second measurement of Q depend on whether it comes before or after a measurement of R, and the two observables do not commute.

In a formal setup, the state of an isolated physical system in quantum mechanics is represented, at a fixed time , by a state vector belonging to a separable complex Hilbert space. Using bra–ket notation the relation between state vector and "position basis" of the Hilbert space can be written as Its relation with an observable can be elucidated by generalizing the quantum state to a measurable function and its domain of definition to a given σ finite measure space This allows for a refinement of Lebesgue's decomposition theorem, decomposing μ into three mutually singular parts

where μac is absolutely continuous with respect to the Lebesgue measure, μsc is singular with respect to the Lebesgue measure and atomless, and μpp is a pure point measure.

Let be atomic (i. e. the set in is an atom); specifying the measure of any discrete variable x ∈ A equal to 1. The amplitudes are composed of state vector indexed by A; its components are denoted by ψ(x) for uniformity with the previous case. If the ℓ^(2) norm of is equal to 1, then is a probability mass function. A convenient configuration space X is such that each point x produces some unique value of the observable Q. For discrete X it means that all elements of the standard basis are eigenvectors of Q. Then is the probability amplitude for the eigenstate If it corresponds to a non degenerate eigenvalue of Q, then gives the probability of the corresponding value of Q for the initial state
if and only if is the same quantum state as 1=ψ(x) = 0 if and only if and are orthogonal. Otherwise the modulus of ψ(x) is between 0 and 1. A discrete probability amplitude may be considered as a fundamental frequency in the probability frequency domain (spherical harmonics) for the purposes of simplifying M theory transformation calculations. Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator.

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. The "transitional" interpretation may be applied to L^(2)s on non discrete spaces as well.