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in the mathematically-rigorous formulation of quantum mechanics developed by paul dirac[8] and john von neumann,[9] the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors"). formally, these reside in a complex separable hilbert space (variously called the "state space" or the "associated hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). in other words, the possible states are points in the projectivization of a hilbert space, usually called the complex projective space. the exact nature of this hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. each observable is represented by a maximally hermitian (precisely: by a self-adjoint) linear operator acting on the state space. each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. if the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues.
in the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function, also referred to as state vector in a complex vector space.[10] this abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. for example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with accuracy. for instance, electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. contours of constant probability, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.[11]
as the result of a measurement, the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate of the observable. the possible results of a measurement are the eigenvalues of the operator representing the observable — which explains the choice of hermitian operators, for which all the eigenvalues are real. we can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.
the probabilistic nature of quantum mechanics thus stems from the act of measurement. this is one of the most difficult aspects of quantum systems to understand. it was the central topic in the famous bohr-einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. in the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. interpretations of quantum mechanics have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the relative state interpretation. the basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. for details, see the article on measurement in quantum mechanics.[12] generally, quantum mechanics does not assign definite values to observables. instead, it makes predictions using probability distributions; that is, the probability of obtaining possible outcomes from measuring an observable. often these results are skewed by many causes, such as dense probability clouds[13] or quantum state nuclear attraction.[14][15] naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. hence, uncertainty is involved in the value. there are, however, certain states that are associated with a definite value of a particular observable. these are known as eigenstates of the observable ("eigen" can be translated from german as inherent or as a characteristic).[16]
in the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. however, quantum mechanics does not pinpoint the exact values of a particle for its position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it only provides a range of probabilities of where that particle might be given its momentum and momentum probability. therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstate). usually, a system will not be in an eigenstate of the observable we are interested in. however, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. this process is known as wavefunction collapse, a debatable process.[17] it involves expanding the system under study to include the measurement device. if one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. for example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. when one measures the position of the particle, it is impossible to predict with certainty the result.[12] it is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. after the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.[18]
the time evolution of a quantum state is described by the schrödinger equation, in which the hamiltonian, the operator corresponding to the total energy of the system, generates time evolution. the time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time.[19]
during a measurement, on the other hand, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random. a time-evolution simulation can be seen here.[20][21] wave functions can change as time progresses. an equation known as the schrödinger equation describes how wave functions change in time, a role similar to newton's second law in classical mechanics. the schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. however, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. this also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates.[22]
some wave functions produce probability distributions that are constant, or independent of time, such as when in a stationary state of constant energy, time drops out of the absolute square of the wave function. many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. for example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (fig. 1). (note that only the lowest angular momentum states, labeled s, are spherically symmetric).[23]
the schrödinger equation acts on the entire probability amplitude, not merely its absolute value. whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states.
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