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Quantum mechanics

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Quantum mechanics or quantum physics is a physical theory formulated in the first half of the twentieth century which successfully describes the behavior of matter at small distance scales.

It explains and quantifies three effects that classical physics cannot account for:

  • 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, classically described purely in terms of discrete particles moving in definite trajectories, 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).

The correspondence principle states that quantum mechanics agrees with the predictions of classical physics for large systems.

Table of contents

Description of the theory

In quantum mechanics, the instantaneous state of a system is described by a wave function which encodes the probability distribution of all measurable properties, or observables. Quantum mechanics only makes predictions about these probability distributions, and does not assign definite values to the observables. The wavelike properties of matter are explained by the interference of wave functions.

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. The time evolution of wave functions is described by the Schrödinger equation.

Some wave functions describe probability distributions which are constant in time. Many systems that would be 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 probability cloud surrounding the nucleus.

When a measurement is performed on an observable of the system, the wavefunction turns into one of a set of wavefunctions called eigenstates of the observable. This is known as wavefunction collapse. The relative probabilities of collapsing into each of the possible eigenstates is described by 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 will collapse into one that is sharply concentrated around the observed position x.

During the process of wavefunction collapse, the wavefunction does not obey the Schrödinger equation. The Schrödinger equation 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. However, during a measurement, the eigenstate to which the wavefunction collapses is probabilistic. The probabilistic nature of quantum mechanics thus stems from the act of measurement.

Mathematical formulation

In the mathematically rigorous formulation developed by Paul Dirac and John von Neumann, a system is described by a complex separable Hilbert space (such as the space of square-integrable "wave"-functions), a state of the system is a unit vector in that space, and every observable is represented by a self-adjoint densely defined linear operator on that 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 probability distribution of an observable in a given state can be computed from the spectral decomposition of the corresponding operator. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. After the measurement is conducted, the system's state will be an eigenstate corresponding to the measured eigenvalue. Heisenberg's uncertainty principle becomes a theorem about non-commuting operators.

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


The original formulation of quantum mechanics was not compatible with special relativity. However, the principles of quantum mechanics can and have been extended into quantum field theories, which are consistent with special relativity. Quantum mechanics as such omits the electromagnetic force, the strong nuclear force, and gravity. The quantum field theory describing electromagnetism is quantum electrodynamics; it is, at least in principle, capable of explaining chemical interactions as well as the interaction of matter and electromagnetic radiation. The quantum field theory describing the strong nuclear force is quantum chromodynamics, which describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force can be unified, in their quantised forms, into a single quantum field theory: electroweak theory. The unification of quantum mechanics with gravity and hence with general relativity has eluded researchers so far (see Theory of everything).


Much of modern technology operates under quantum mechanical principles. Examples include the laser, 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 has 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.

Philosophical debate

Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate.

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. 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.

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.


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.

In the 1940s, quantum electrodynamics was developed by Feynman, Dyson, Schwinger[?], and Tomonaga. It served as a role model for subsequent quantum field theories.

The many worlds interpretation was formulated by Everett[?] in 1956.

Quantum chromodynamics had a long history, beginning 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.

Some quotations

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 Bohr

God does not play dice with the cosmos.
Albert Einstein

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

It's always fun to learn something new about quantum mechanics.
Benjamin Schumacher[?]

External Links

  • A history of quantum mechanics (http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_Quantum_age_begins)
  • George W Mackey, "The mathematical foundations of quantum mechanics", New York, W. A. Benjamin, 1963.

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