# Quantum mechanics

(increasing across) and energy (increasing down), the electron orbitals of a hydrogen atom can be found in different locations. The angular momentum and energy increase only in these quantum steps.]]

**Quantum mechanics**is a physical theory which at very small distances produces results that are very different and much more accurate than the results of classical mechanics. It is the underlying framework of many fields of physics and chemistry, including condensed matter physics, quantum chemistry, and particle physics. It is derived from a small set of basic principles, and applies to at least three general types of phenomena that classical mechanics and classical electrodynamics cannot account for: quantization, wave-particle duality (interference of matter particles), and quantum entanglement. Additionally, there are countless more specific systems where quantum mechanical predictions differ from the classical theory. For example, according to classical mechanics stable atoms cannot exist and the total radiation of a black body is infinite.

The terms **quantum physics** and **quantum theory** are often used as synonyms of quantum mechanics. Some authors refer to "quantum mechanics" in the restricted sense of non-relativistic quantum mechanics, as opposed to quantum field theory. This meaning shall not be used in this article; we will take "quantum mechanics" to mean quantum theory in its most general sense.

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Paul Dirac, Richard Feynman and others. Some aspects of the theory are still actively studied.

Table of contents |

2 Applications 3 Mathematical formulation 4 Interactions with other theories of physics 5 Philosophical consequences 6 History 7 References 8 See also |

## Description of the theory

### Wave functions and measurement

There are a number of mathematically equivalent formulations of quantum mechanics. One of these formulations was developed by Erwin Schrödinger in which quantum mechanics describes the instantaneous state of a system with a wave function that encodes the probability distribution of all measurable properties, or observables. Possible observables for a system include energy, position, momentum, and angular momentum. Quantum mechanics does not assign definite values to the observables; instead, it makes predictions about their probability distributions, which is calculated by applying a mathematical operator to the wave function. From the mathematical operator, one can find the possible observed values of the observables. In some cases the observables can only have discrete values. Such observables are said to be quantized.

In Schrödinger's formulation, wave functions can change as time progresses. For example, a particle moving in empty space may be described by a wave function that is a wave packet centered around some mean position. As time progresses, the center of the wave packet changes, so that the particle becomes more likely to be located at a different position. On the other hand, some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, an electron in an unexcited atom is pictured classically as a particle circling the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric probability cloud surrounding the nucleus.

The act of measurement of an observable of the system will result in the modification of the system and its wave function. Immediately after a measurement is performed, the wavefunction becomes one of the wavefunctions compatible with the measurement, i.e. a wavefunction that gives 100% probability for the result obtained. This process is known as wavefunction collapse. (See relative state interpretation for an alternative to this. The conversion of a pure state into a mixed state only occurs when we take the partial trace and is due to quantum entanglement with the environment, and not the process of measurement per se.) The probability of collapsing into a given wave function depends on the type of measurement, and can be computed from the instantaneous wavefunction just before the collapse. Consider the above example of a particle moving in empty space. If we measure the particle's position, we will obtain a random value *x*. In general, it is impossible for us to predict with certainty the value of *x* which we will obtain, although it is probable that we will obtain one that is near the center of the wave packet, where the amplitude of the wave function is large. After the measurement has been performed, the wavefunction of the particle collapses into one that is sharply concentrated around the observed position *x*. The measurement of the speed of the particle would result in a totally different wave function.

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. During a measurement, the change of the wavefunction into another one is probabilistic, not deterministic. The probabilistic nature of quantum mechanics thus stems from the act of measurement.

### Quantum mechanical effects

The first type of quantum effect is the quantization of certain physical quantities. We have seen that certain observables in quantum mechanics take on discrete rather than continuous values. Examples of quantized observables include angular momentum, the total energy of a bound system, and the energy contained in an electromagnetic wave of a given frequency. It should be noted that not all observables are quantized; for example, the position of a particle is a continuous observable.

Another quantum effect is the uncertainty principle, which is the phenomenon that consecutive measurements of two or more observables may possess a fundamental limitation on accuracy. For example, the position and the momentum of a particle can never be simultaneously measured with arbitrary precision, even in principle: as the precision of the position measurement improves, the maximum precision of the momentum measurement decreases, and vice versa. Those variables for which it holds (e.g. momentum and position, or energy and time) are canonically conjugate variables in classical physics.

Another quantum effect is the wave-particle duality. It has been shown that, under certain experimental conditions, microscopic objects like atoms or electrons exhibit particle-like behavior, such as scattering. ("Particle-like" in the sense of an object that can be localized to a particular region of space.) Under other conditions, the same type of objects exhibit wave-like behavior, such as interference. We can observe only one type of property at a time.

Another quantum effect is quantum entanglement. In some cases, the wave function of a system composed of many particles cannot be separated into independent wave functions, one for each particle. In that case, the particles are said to be entangled. Entangled particles display remarkable and counter-intuitive properties. For example, a measurement made on a particle can produce, through the collapse of the total wavefunction, an instantaneous effect on the other particles with which it is entangled, even if they are far away.

## Applications

Much of modern technology operates at a scale where quantum effects are significant. Examples include the laser, the transistor, the electron microscope, and magnetic resonance imaging. Most of the calculations performed in computational chemistry rely on quantum mechanics.

Many of the phenomena studied in condensed matter physics are fully quantum mechanical, and cannot be satisfactorily modeled using classical physics. This includes the electronic properties of solids, such as superconductivity and semiconductivity. The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics.

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks with much greater efficiency than classical computers. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances.

## Mathematical formulation

In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called *state vectors*) residing in a complex separable Hilbert space (called the *state space*.) The exact nature of the 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 electron is just the product of two complex planes. 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 the time evolution.

Each observable is represented by a densely-defined Hermitian 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. During a measurement, the probability that a system collapses to each eigenstate is given by the absolute square of the inner product between the eigenstate vector and the state vector just before the measurement. The possible results of a measurement are the eigenvalues of the operator - which explains the choice of *Hermitian* operators -- all their eigenvalues are real. We can therefore 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 details of the mathematical formulation are contained in the article Mathematical formulation of quantum mechanics. A discussion of foundations of quantum mechanics is contained in the article on Quantum logic.

## Interactions with other theories of physics

The fundamental rules of quantum mechanics are very broad. They state that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators. These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical (i.e. non-quantum) physics when a system becomes large, which is known as the *classical* or *correspondence limit*. One may therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit.

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field rather than a fixed set of particles. The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.

The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical *1/r* Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory.

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity.

Semi-classical approximations are techniques that make it possible to formulate a quantum problem with some physical quantities replaced by their classical analogues, in an effort to reduce the complexity of the model. Even within non-relativistic quantum mechanics, a fully microscopic treatment generally requires large-scale numerical computations. Analytic quantum solutions that describe the system behavior in terms of known mathematical functions are available only for a small class of systems, of which the harmonic oscillator and the hydrogen atom are the most important representatives.

Even the helium atom, containing just one more electron than hydrogen, defies all attempts at a fully analytic treatment in quantum mechanics. In such a situation, approximate semi-classical results can provide valuable insights. The necessary methods rely on a detailed understanding of the corresponding classical mechanics, allowing in particular for the existence of chaos. The study of these approximations belongs to the field of quantum chaos.

## Philosophical consequences

Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpretations. See interpretation of quantum mechanics for more detail.

The Copenhagen interpretation, due largely to Niels Bohr, was the standard interpretation of quantum mechanics when it was first formulated. According to it, the probabilistic nature of quantum mechanics predictions cannot be explained in terms of some other deterministic theory, and do not simply reflect our limited knowledge. Quantum mechanics provides probabilistic results because the physical universe is itself probabilistic rather than deterministic.

Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. He held that quantum mechanics must be incomplete, and produced a series of objections to the theory. The most famous of these was the EPR paradox. John Stewart Bell showed that the EPR paradox led to experimentally testable differences between quantum mechanics and local hidden variable theories. Experiments have been taken as confirming that quantum mechanics is correct and the real world cannot be described in terms of such hidden variables. "Loopholes" in the experiments, however, mean the question is still not quite settled.

The many worlds interpretation, formulated in 1956, 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 non-deterministic behavior governed by probabilities because we can observe only the universe we inhabit.

The Bohm interpretation postulates the existence of a non-local, universal wavefunction (Schrödinger equation) which allows distant particles to interact instantaneously. It is not popular among physicists largely because it is considered very inelegant.

## History

In 1900, Max Planck introduced the idea that energy is quantized, in order to derive a formula for the observed frequency dependence of the energy emitted by a black body. In 1905, Einstein explained the photoelectric effect by postulating that light energy comes in quanta called photons. In 1913, Bohr explained the spectral lines of the hydrogen atom, again by using quantization. In 1924, Louis de Broglie put forward his theory of matter waves.

These theories, though successful, were strictly phenomenological: there was no rigorous justification for quantization. They are collectively known as the *old quantum theory*.

The phrase "quantum physics" was first used in Johnston's *Planck's Universe in Light of Modern Physics*.

Modern quantum mechanics was born in 1925, when Heisenberg developed matrix mechanics and Schrödinger invented 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. In 1927, Paul Dirac 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.

The field of quantum chemistry was pioneered by Walter Heitler and Fritz London, who published a study of the covalent bond of the hydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large number of workers, including the American chemist Linus Pauling.

In the 1940s, quantum electrodynamics was developed by Feynman, Dyson, Schwinger, and Tomonaga. It served as a role model for subsequent quantum field theories. See: Feynman's path integral formulation of Quantum mechanics.

The many worlds interpretation was formulated by Everett in 1956.

Quantum chromodynamics began in the early 1960s. The theory as we know it today was formulated by Polizter, Gross and Wilzcek in 1975. Building on pioneering work by Schwinger, Higgs, Goldstone and others, Glashow, Weinberg and Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force.

Recently, there has been much interest in quantum information.

## References

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

## See also

- Quantum mechanics
- A history of quantum mechanics
- David Mermin on the future directions of physics
- - new developments in the understanding of the quantum-classical relation

General subfields within physics
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Classical mechanics | Condensed matter physics | Continuum mechanics | Electromagnetism | General relativity | Particle physics | Quantum field theory | Quantum mechanics | Solid state physics | Special relativity | Statistical mechanics | Thermodynamics |