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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.

Physical overview

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.

Mathematical formulation

In a formal setup, the state of an isolated physical system in quantum mechanics is represented, at a fixed time ${\displaystyle t}$, by a state vector |Ψ⟩ belonging to a separable complex Hilbert space. Using bra–ket notation the relation between state vector and "position basis" ${\displaystyle \{|x\rangle \}}$ of the Hilbert space can be written as^[1]

${\displaystyle \psi (x)=\langle x|\Psi \rangle }$.

Its relation with an observable can be elucidated by generalizing the quantum state ${\displaystyle \psi }$ to a measurable function and its domain of definition to a given σ-finite measure space ${\displaystyle (X,{\mathcal {A}},\mu )}$. This allows for a refinement of Lebesgue's decomposition theorem, decomposing μ into three mutually singular parts

${\displaystyle \mu =\mu _{\mathrm {ac} }+\mu _{\mathrm {sc} }+\mu _{\mathrm {pp} }}$

where μ_ac is absolutely continuous with respect to the Lebesgue measure, μ_sc is singular with respect to the Lebesgue measure and atomless, and μ_pp is a pure point measure.^[2][3]

Continuous amplitudes

A usual presentation of the probability amplitude is that of a wave function ${\displaystyle \psi }$ belonging to the L² space of (equivalence classes of) square integrable functions, i.e., ${\displaystyle \psi }$ belongs to L²(X) if and only if

${\displaystyle \|\psi \|^{2}=\int _{X}|\psi (x)|^{2}\,dx<\infty }$.

If the norm is equal to 1 and ${\displaystyle |\psi (x)|^{2}\in \mathbb {R} _{\geq 0}}$ such that

${\displaystyle \int _{X}|\psi (x)|^{2}\,dx\equiv \int _{X}\,d\mu _{ac}(x)=1}$,

then ${\displaystyle |\psi (x)|^{2}}$ is the probability density function for a measurement of the particle's position at a given time, defined as the Radon–Nikodym derivative with respect to the Lebesgue measure (e.g. on the set R of all real numbers). As probability is a dimensionless quantity, |ψ(x)|² must have the inverse dimension of the variable of integration x. For example, the above amplitude has dimension , where L represents length.

Whereas a Hilbert space is separable if and only if it admits a countable orthonormal basis, the range of a continuous random variable ${\displaystyle x}$ is an uncountable set (i.e. the probability that the system is "at position ${\displaystyle x}$" will always be zero). As such, eigenstates of an observable need not necessarily be measurable functions belonging to L²(X) (see normalization condition below). A typical example is the position operator ${\displaystyle {\hat {\mathrm {x} }}}$ defined as

${\displaystyle \langle x|{\hat {\mathrm {x} }}|\Psi \rangle ={\hat {\mathrm {x} }}\langle x|\Psi \rangle =x_{0}\psi (x),\quad x\in \mathbb {R} ,}$

whose eigenfunctions are Dirac delta functions

${\displaystyle \psi (x)=\delta (x-x_{0})}$

which clearly do not belong to L²(X). By replacing the state space by a suitable rigged Hilbert space, however, the rigorous notion of eigenstates from spectral theorem as well as spectral decomposition is preserved.^[4]

Discrete amplitudes

Let ${\displaystyle \mu _{pp}}$ be atomic (i.e. the set ${\displaystyle A\subset X}$ in ${\displaystyle {\mathcal {A}}}$ is an atom); specifying the measure of any discrete variable x ∈ A equal to 1. The amplitudes are composed of state vector |Ψ⟩ indexed by A; its components are denoted by ψ(x) for uniformity with the previous case. If the ℓ²-norm of |Ψ⟩ is equal to 1, then |ψ(x)|² 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 ${\displaystyle \psi (x)}$ is the probability amplitude for the eigenstate |x⟩. If it corresponds to a non-degenerate eigenvalue of Q, then ${\displaystyle |\psi (x)|^{2}}$ gives the probability of the corresponding value of Q for the initial state |Ψ⟩.

|ψ(x)| = 1 if and only if |x⟩ is the same quantum state as |Ψ⟩. ψ(x) = 0 if and only if |x⟩ and |Ψ⟩ 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.^{[citation needed]} Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator.^{[clarification needed]}

Examples

An example of the discrete case is a quantum system that can be in two possible states, e.g. the polarization of a photon. When the polarization is measured, it could be the horizontal state ${\displaystyle |H\rangle }$ or the vertical state ${\displaystyle |V\rangle }$. Until its polarization is measured the photon can be in a superposition of both these states, so its state ${\displaystyle |\psi \rangle }$ could be written as

${\displaystyle |\psi \rangle =\alpha |H\rangle +\beta |V\rangle }$,

with ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ the probability amplitudes for the states ${\displaystyle |H\rangle }$ and ${\displaystyle |V\rangle }$ respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is ${\displaystyle |\alpha |^{2}}$, and the probability of being vertically polarized is ${\displaystyle |\beta |^{2}}$.

Hence, a photon in a state ${\textstyle |\psi \rangle ={\sqrt {\frac {1}{3}}}|H\rangle -i{\sqrt {\frac {2}{3}}}|V\rangle }$ would have a probability of ${\textstyle {\frac {1}{3}}}$ to come out horizontally polarized, and a probability of ${\textstyle {\frac {2}{3}}}$ to come out vertically polarized when an ensemble of measurements are made. The order of such results, is, however, completely random.

Another example is quantum spin. If a spin-measuring apparatus is pointing along the z-axis and is therefore able to measure the z-component of the spin (${\mathrm{\sigma <!--\; \sigma \; -->}}_z$Zdroj:https://en.wikipedia.org?pojem=Probability_amplitude
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