Part of a series of articles about
Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Local Superdeterminism Many-worlds Objective collapse Quantum logic Relational Transactional Von Neumann–Wigner
Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos EPR paradox Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning
Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose de Broglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Lamb Landau Laue Moseley Millikan Onnes Pauli Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger
v t e

Quantum mechanics is a fundamental theory in physics that describes the behavior of nature at and below the scale of atoms.^[2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.

Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic/microscopic) scale.^[3]

Quantum systems have bound states that are quantized to discrete values of energy, momentum, angular momentum, and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves (wave–particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).

Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper, which explained the photoelectric effect. These early attempts to understand microscopic phenomena, now known as the "old quantum theory", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born, Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle's energy, momentum, and other physical properties may yield.

Overview and fundamental concepts

Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms,^[4] but its application to human beings raises philosophical problems, such as Wigner's friend, and its application to the universe as a whole remains speculative.^[5] Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy. For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10¹² when predicting the magnetic properties of an electron.^[6]

A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number, known as a probability amplitude. This is known as the Born rule, named after physicist Max Born. For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.^{[7]: 67–87}

One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between different measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum.^{[7]: 427–435}

Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.^{[8]: 102–111 [2]: 1.1–1.8} The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.^[8] However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave).^{[8]: 109 [9][10]} However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as wave–particle duality. In addition to light, electrons, atoms, and molecules are all found to exhibit the same dual behavior when fired towards a double slit.^[2]

Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling: a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential.^[11] In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay, nuclear fusion in stars, and applications such as scanning tunnelling microscopy,tunnel diode and tunnel field-effect transistor.^[12][13]

When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".^[14] Quantum entanglement enables quantum computing and is part of quantum communication protocols, such as quantum key distribution and superdense coding.^[15] Contrary to popular misconception, entanglement does not allow sending signals faster than light, as demonstrated by the no-communication theorem.^[15]

Another possibility opened by entanglement is testing for "hidden variables", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory can provide. A collection of results, most significantly Bell's theorem, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then the results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.^[16][17]

It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra, differential equations, group theory, and other more advanced subjects.^[18][19] Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.

Mathematical formulation

In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ${\displaystyle \psi }$ belonging to a (separable) complex Hilbert space ${\displaystyle {\mathcal {H}}}$. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys ${\displaystyle \langle \psi ,\psi \rangle =1}$, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ${\displaystyle \psi }$ and ${\displaystyle e^{i\alpha }\psi }$ represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions ${\displaystyle L^{2}(\mathbb {C} )}$, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors ${\displaystyle \mathbb {C} ^{2}}$ with the usual inner product.

Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue ${\displaystyle \lambda }$ is non-degenerate and the probability is given by ${\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}$, where ${\displaystyle {\vec {\lambda }}}$ is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ${\displaystyle \langle \psi ,P_{\lambda }\psi \rangle }$, where ${\displaystyle P_{\lambda }}$ is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the probability density.

After the measurement, if result ${\displaystyle \lambda }$ was obtained, the quantum state is postulated to collapse to ${\displaystyle {\vec {\lambda }}}$, in the non-degenerate case, or to ${\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}$, in the general case. 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. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wave function collapse" (see, for example, the many-worlds interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases to exist as an independent entity (see measurement in quantum mechanics^[20]).

Time evolution of a quantum state

The time evolution of a quantum state is described by the Schrödinger equation:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (t)=H\psi (t).}$

Here ${\displaystyle H}$ denotes the Hamiltonian, the observable corresponding to the total energy of the system, and ${\displaystyle \hbar }$ is the reduced Planck constant. The constant ${\displaystyle i\hbar }$ is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle.

The solution of this differential equation is given by

${\displaystyle \mathrm{\psi <!--\; \psi \; -->}(t)=e^{-<!--\; -\; -->iHt/\mathrm{\hslash <!--\; \hslash \; -->}}\mathrm{\psi <!--\; \psi \; -->}(}$Zdroj:https://en.wikipedia.org?pojem=Quantum_physics
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