- The values of some measurable variables of a system, most notably the total energy of a bounded system, can attain only certain discrete values determined by the system. (The smallest possible jumps in the values of those observables are called "quanta" (Latin
*quantum*, quantity), hence the name quantum mechanics.) - Matter exhibits properties of waves (see wave-particle duality).
- Certain pairs of observables, for example the position and momentum of a particle, can never be simultaneously ascertained to arbitrary precision (see Heisenberg's uncertainty principle).

In quantum mechanics, all of these are resolved by describing the instantaneous state of a system with a *wave function* that encodes the probability distributions of all observables.
Quantum mechanics makes predictions only about these probability distributions and not about the precise values of observables.
The wavelike nature of matter is readily explained as interference effects between probability waves.
Many systems that were formerly seen as changing over time (for instance, an electron circling a proton) are now described as static (a proton surrounded by a "probability cloud" describing the likelihood of locating the electron at a specific place).
If the probability distributions do change over time, then the Schrödinger equation is used to describe the corresponding evolution of the wave function.

In the formal mathematical theory, the state of a system is described by an element of a Hilbert space and observables are modeled as self-adjoint operators on this Hilbert space. Given a state and such an operator, the probabilities for the various outcomes of the corresponding observation can be calculated. The time evolution of a system is described by the Schrödinger equation, in which the Hamiltonian?, the operator corresponding to the energy observable, plays a prominent role.

The details of the mathematical formulation are contained in the article mathematical formulation of quantum mechanics.

Quantum mechanics omits the weak nuclear force, the strong nuclear force, gravity and a full treatment of the the electromagnetic force. The principles of quantum mechanics can be applied to the well-established classical field theories. If electromagnetism is quantized, the resulting quantum field theory is called quantum electrodynamics. It is (at least in principle) able to explain the chemical elements and molecules and their properties, as well as interactions of matter and electromagnetic radiation. If the strong nuclear force is quantized, one obtains quantum chromodynamics, which describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force can be combined, in their quantised forms, and shown to be expressions of one underlying quantum field theory: [electroweak theory]?. The unification of these theories with gravity and hence with general relativity has so far eluded researchers (see Theory of everything).

Quantum mechanical explanations for the behavior of transistors and diodes underlie all of modern technology. Quantum mechanics has been used to build lasers and electron microscopes as well as nuclear magnetic resonance imaging, which is also known as magnetic resonance imaging when used in medicine. Computational chemistry includes applied quantum mechanics done on a computer. There are efforts underway to build quantum computers which process vast amounts of data by exploiting the possibility of one system being in several states at once.

Quantum mechanics has provoked a strong philosophical debate. The fundamental problem is that causality and determinism is lost: while the probability distributions evolve according to a well-established deterministic law, the values of the observables themselves do not. Because of this, Albert Einstein held that quantum mechanics must be incomplete. He rejected the now standard Copenhagen interpretation by Niels Bohr which contends that quantum theory describes all there is to know about reality and that the probability statements are irreducible and do not simply reflect our limited knowledge. This interpretation further holds that the act of observation overrides the Schrödinger equation and causes the system to instantaneously change to an eigenstate (the so-called collapse of the wave function). Everett's newer many worlds interpretation holds that all the possibilities described by quantum theory simultaneously occur in a "multiverse" composed of mostly independent parallel universes. While the multiverse is deterministic, we perceive nondeterministic behavior governed by probabilities because we can observe only one "random" universe, with other copies of ourselves observing others. In Everett's interpretation, the act of observation is described by the regular Schrödinger equation and does not require a special treatment.

While attempting to derive the correct frequency dependence of the energy emitted by a black body at a certain temperature, Max Planck in 1900 introduced the idea that energy is quantized. In 1905, Einstein explained the photoelectric effect by postulating that light energy comes in quanta called photons. In 1913, Bohr deduced the spectral lines of the hydrogen atom again by using quantization. [Louis de Broglie]? put forward his theory of matter-as-wave in 1924. Starting in 1925, Heisenberg developed his matrix method, while Schrödinger introduced wave mechanics and the Schrödinger equation. Schrödinger subsequently showed that the two approaches were equivalent. Heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation took shape at about the same time. Paul Dirac, in 1928, unified quantum mechanics with special relativity. He also pioneered the use of operator theory, including the influential bra-ket notation. In 1932, John von Neumann formulated the rigorous mathematical basis for quantum mechanics as operator theory. Quantum electrodynamics was fully developed by Feynman, Dyson, Schwinger and Tomonaga starting in the late 1940s, and served as a role model for subsequent quantum field theories. The many worlds interpretation was formulated by Hugh Everett III in 1956. Quantum chromodynamics was proposed in 1964 by Greenberg and Nambu.

*I do not like it, and I am sorry I ever had anything to do with it.*- Erwin Schrödinger, speaking of quantum mechanics

*Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it.*- Niels Henrik David Bohr

*God does not play dice with the universe.*- Albert Einstein

*I think it is safe to say that no one understands quantum mechanics.*- Richard Feynman

- [A history of quantum mechanics]
- George W Mackey, "The mathematical foundations of quantum mechanics", New York, W. A. Benjamin, 1963.

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