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Quantum mechanics is a physical theory that describes the behavior of physical systems at short distances. Quantum
mechanics provides a mathematical framework derived from a small set of basic principles capable of producing
experimental predictions for three types of phenomena that classical mechanics and classical electrodynamics cannot account for: quantization, wave-particle
duality, and quantum entanglement. The related terms
quantum physics and quantum theory are sometimes used as synonyms of quantum mechanics, but
also to denote a superset of theories, including pre-quantum mechanics old quantum theory (see #History), or, when the term quantum mechanics is used in a more restricted sense, to include theories like quantum field theory.
Quantum mechanics is the underlying theory of many fields of physics and chemistry, including condensed matter
physics, quantum chemistry, and particle physics.
Quantization
Main article: Quantization
(physics).
Quantization is the taking of discrete rather than continuous values for some physical quantities (e.g.
the total energy of a black body).
When a quantity can only take on integer multiples of some base value, the smallest possible intervals between the discrete
values are quanta. The size of the quanta typically varies from system to system.
Wave-particle duality
Under certain experimental conditions, microscopic objects like atoms or electrons exhibit wave-like behavior, such as
interference. Under other conditions, the same species of objects
exhibit particle-like behavior ("particle" meaning an object that can be localized to a particular region of space), such as scattering. This phenomenon is
known as wave-particle duality.
Quantum entanglement
Physical properties of objects with related histories may be correlated to an extent forbidden by any classical theory, to an
extent that they may only be accurately described with reference to each other. This phenomenon is called entanglement and its difference from ordinary correlation is
described by Bell's inequality. Measurements of violations of
Bell's inequality were one of the major verifications of the
quantum theory.
Description of the theory
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 making
predictions about their probability distributions.
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 that 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, spherically symmetric 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 that
are called eigenstates of the observable. This process 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 collapses into one that is sharply
concentrated around the observed position x.
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. During a
measurement, the eigenstate to which the wavefunction collapses is probabilistic, not deterministic. 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, 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
relativistic 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 quantum
chromodynamics, which describes the interactions of the subnuclear particles, the 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.
Applications
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. Another
active research topic is quantum teleportation, which
deals with techniques to transmit quantum states over arbitrary distances.
Philosophical debate
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's theoretical solution to the EPR paradox, and
its later experimental verification, disproved a large class of such hidden variable theories and persuaded the majority of
physicists that quantum mechanics is not an approximation to a nominally classical hidden-variable theory.
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
The foundations of quantum mechanics were established during the first half of the 20th century by the work of Max Planck, Niels Bohr, Werner
Heisenberg, Erwin Schrödinger, Paul Dirac, and others. Some fundamental aspects of the theory are still being actively studied.
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. See: Feynman's path integral formulation of Quantum mechanics.
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.
Recently, there has been much interest in quantum
information.
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
- "Who are you to tell God what to do?"
- Niels Bohr in response to 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
- "If that turns out to be true, I'll quit physics."
- Max von Laue, Nobel Laureate 1914, of de Broglie's thesis on electrons
having wave properties.
- "Anyone wanting to discuss a quantum mechanical problem had better understand and learn to apply quantum mechanics to
that problem."
- Willis Lamb, Nobel Laureate 1955
Reference
- George W Mackey, "The mathematical foundations of quantum mechanics", New York, W. A. Benjamin, 1963.
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