Quantum mechanics


Quantum mechanics, also referred to as quantum physics, is a physical theory
that describes the behavior of matter at short length scales. It provides a
quantitative explanation for two types of phenomena that classical mechanics
and classical electrodynamics cannot account for:

   * Some observable physical quantities, such as the total energy of a
     blackbody, take on discrete rather than continuous values. This
     phenomenon is called quantization, and the smallest possible intervals
     between the discrete values are called quanta (singular: quantum, from
     the Latin word for "quantity", hence the name "quantum mechanics.") The
     size of the quanta typically varies from system to system.
   * Under certain experimental conditions, microscopic objects like atoms
     and 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.

The foundations of quantum mechanics were established during the first half
of the 20th century by the work of Niels Bohr, Werner Heisenberg, Erwin
Schršdinger, Paul Dirac, and others. Some fundamental aspects of the theory
are still being actively studied. Quantum mechanics has also been adopted as
the underlying theory of many fields of physics and chemistry, including
condensed matter physics, quantum chemistry, and particle physics.

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 Schršdinger 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.

During the process of wavefunction collapse, the wavefunction does not obey
the Schršdinger equation. The Schršdinger 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.

One of the consequences of wavefunction collapse is that certain pairs of
observables, such as position and momentum, can never be simultaneously
ascertained to arbitrary precision. This effect is known as Heisenberg's
uncertainty principle.

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.
The time evolution of a quantum state is described by the Schršdinger
equation, in which the Hamiltonian, the operator corresponding to the total
energy of the system, plays a central role.

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. 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.

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 Schršdinger 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.

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.

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.

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 Schršdinger invented wave mechanics and the Schršdinger
equation. Schršdinger 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.

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 Schršdinger, 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

     Einstein, don't 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