Quantum Theory
Quantum mechanics is a fundamental branch of theoretical
physics that replaces Newtonian mechanics and classical electromagnetism at the
atomic and subatomic levels. It is the underlying mathematical framework of
many fields of physics and chemistry, including condensed matter physics,
atomic physics, molecular physics, computational chemistry, quantum chemistry,
particle physics, and nuclear physics. Along with general relativity, quantum
mechanics is one of the pillars of modern physics.
Introduction
The term quantum
(Latin, "how much") refers to discrete units that the theory assigns
to certain physical quantities, such as the energy of an atom at rest (see
Figure 1, at right). The discovery that waves could be measured in
particle-like small packets of energy called quanta led to the branch of
physics that deals with atomic and subatomic systems which we today call
Quantum Mechanics.
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 Schrodinger,
Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory
are still actively studied.
Quantum mechanics is a more fundamental theory than
Newtonian mechanics and classical electromagnetism, in the sense that it
provides accurate and precise descriptions for many phenomena that these
"classical" theories simply cannot explain on the atomic and
subatomic level. It is necessary to use quantum mechanics to understand the
behavior of systems at atomic length scales and smaller. For example, if
Newtonian mechanics governed the workings of an atom, electrons would rapidly
travel towards and collide with the nucleus. However, in the natural world the
electron normally remains in a stable orbit around a nucleus -- seemingly
defying classical electromagnetism.
Quantum mechanics was initially developed to explain the
atom, especially the spectra of light emitted by different atomic species. The
quantum theory of the atom developed as an explanation for the electron's
staying in its orbital, which could not be explained by
Quantum mechanics uses complex number wave functions
(sometimes referred to as orbitals in the case of
atomic electrons), and more generally, elements of a complex vector space to
explain such effects. These are related to classical physics largely through
probability.
Probability in the context of quantum mechanics has to do
with the likelihood of finding a system in a particular state at a certain
time, for example, finding an electron, in a particular region around the
nucleus at a particular time. Therefore, electrons cannot be pictured as
localized particles in space but rather should be thought of as
"clouds" of negative charge spread out over the entire orbit.
These clouds represent the regions around the nucleus where
the probability of "finding" an electron is the largest. This
probability cloud obeys a quantum mechanical principle called Heisenberg's
Uncertainty Principle, which states that there is an uncertainty in the classical
position of any subatomic particle, including the electron; so instead of
describing where an electron or other particle is, the entire range of possible
values is used, describing a probability distribution.
So in normal atoms with electrons in stationary states, the
probability of the electron being within the nucleus (or somewhere else in atom
within similarly small volume) is nearly zero according to the Uncertainty Principle
(it is nearly zero as the nucleus has a volume and is not a point). Therefore,
quantum mechanics, translated to
The other exemplar that led to quantum mechanics was the study
of electromagnetic waves such as light. When it was found in 1900 by Max Planck
that the energy of waves could be described as consisting of small packets or
quanta, Albert Einstein exploited this idea to show that an electromagnetic
wave such as light could be described by a particle called the photon with a
discrete energy dependent on its frequency.
This led to a theory of unity between subatomic particles
and electromagnetic waves called wave-particle duality in which particles and
waves were neither one nor the other, but had certain properties of both. While
quantum mechanics describes the world of the very small, it also is needed to
explain certain "macroscopic quantum systems" such as superconductors
and superfluids.
Broadly speaking, quantum mechanics incorporates four
classes of phenomena that classical physics cannot account for:
- the quantization (discretization) of certain physical quantities,
- wave-particle duality,
- the uncertainty
principle,
- quantum
entanglement.
Each of these phenomena will be described in greater detail
in subsequent sections.
Most physicists believe that quantum mechanics provides a
correct description for the physical world under almost all circumstances.
However, the effects of quantum mechanics are generally not significant when
considering the observable Universe as a whole. This is because although atoms
and subatomic particles are the building blocks of matter, when analyzing the
universe on large scales one finds that the dominant force becomes gravity -- which
is described using Einstein's general theory of relativity. In some cases, both
general relativity and quantum mechanics converge. As an example, general
relativity is unable to explain what will happen if a subatomic particle hits
the singularity of a black hole which is a phenomenon predicted by general
relativity and involves gravity in the macro world. Only quantum mechanics can
provide the answer: the particle's position will have an uncertainty that
follows the Heisenberg Uncertainty Principle, such that it might not really
reach the singularity and thus escape the possible collapse to infinite
density.
It is believed that the theories of general relativity and
quantum mechanics, the two great achievements of physics in the 20th century,
contradict one another for two main reasons.
One is that the former is an essentially deterministic
theory and the latter is essentially indeterministic.
Secondly, general relativity relies mainly on the force of
gravity while quantum mechanics relies mainly on the other three fundamental
forces, those being the strong, the weak, and the electromagnetic.
The question of how to resolve this contradiction remains
an area of active research (see, for example, quantum gravity).In certain
situations, the laws of classical physics approximate the laws of quantum
mechanics to a high degree of precision.
This is often expressed by saying that in case of large
quantum numbers quantum mechanics "reduces" to classical mechanics
and classical electromagnetism . This situation is
called the correspondence, or classical limit.
Quantum mechanics can be formulated in either a
relativistic or non-relativistic manner. Relativistic quantum mechanics
(quantum field theory) provides the framework for some of the most accurate
physical theories known.
Still, non-relativistic quantum mechanics is also used due
to its simplicity and when relativistic effects are relatively small. We will
use the terms quantum mechanics, quantum physics, and quantum theory
synonymously, to refer to both relativistic and non-relativistic quantum
mechanics.
It should be noted, however, that certain authors refer to
"quantum mechanics" in the more restricted sense of non-relativistic
quantum mechanics. Also, in quantum mechanics, the use of the term particle
refers to an elementary or subatomic particle.
Description of the Theory
There are a number of mathematically equivalent
formulations of quantum mechanics. One of the oldest and most commonly used
formulations is the transformation theory invented by
In this formulation, the instantaneous state of a quantum
system encodes the probabilities of its measurable properties, or
"observables". Examples of observables include energy, position,
momentum, and angular momentum. Observables can be either continuous (e.g., the
position of a particle) or discrete (e.g., the energy of an electron bound to a
hydrogen atom).
Generally, quantum mechanics does not assign definite
values to observables. Instead, it makes predictions about probability
distributions; that is, the probability of obtaining each of the possible
outcomes from measuring an observable.
Naturally, these probabilities will depend on the quantum
state at the instant of the measurement. There are, however, certain states
that are associated with a definite value of a particular observable. These are
known as "eigenstates" of the observable
("eigen" meaning "own" in
German). In the everyday world, it is natural and intuitive to think of
everything being in an eigenstate of every
observable. Everything appears to have a definite position, a definite
momentum, and a definite time of occurrence.
However, Quantum Mechanics does not pinpoint the exact
values for the position or momentum of a certain particle in a given space in a
finite time, but, rather, it only provides a range of probabilities of where
that particle might be. Therefore, it became necessary to use different words
for a) the state of something having an uncertainty relation and b) a state
that has a definite value.
The latter is called the "eigenstate"
of the property being measured.A concrete example
will be useful here. Let us consider a free particle.
In quantum mechanics, there is wave-particle duality so the
properties of the particle can be described as a wave. Therefore, its quantum
state can be represented as a wave, of arbitrary shape and extending over all
of space, called a wavefunction.
The position and momentum of the particle are observables.
The Uncertainty Principle of quantum mechanics states that both the position
and the momentum cannot simultaneously be known with infinite precision at the
same time.
However, we can measure just the position alone of a moving
free particle creating an eigenstate of position with
a wavefunction that is very large at a particular
position x, and zero everywhere else.
If we perform a position measurement on such a wavefunction, we will obtain the result x with 100%
probability.
In other words, we will know the position of the free
particle. This is called an eigenstate of position.
If the particle is in an eigenstate of position then
its momentum is completely unknown. An eigenstate of
momentum, on the other hand, has the form of a plane wave. It can be shown that
the wavelength is equal to h/p, where h is Planck's constant and p is the
momentum of the eigenstate.
If the particle is in an eigenstate
of momentum then its position is completely blurred out.Usually,
a system will not be in an eigenstate of whatever
observable we are interested in. However, if we measure the observable, the wavefunction will immediately become an eigenstate
of that observable.
This process is known as wavefunction
collapse. If we know the wavefunction at the instant
before the measurement, we will be able to compute the probability of
collapsing into each of the possible eigenstates.
For example, the free particle in our previous example will
usually have a wavefunction that is a wave packet
centered around some mean position x0, neither an eigenstate of position nor of momentum. When we measure the
position of the particle, it is impossible for us to predict with certainty the
result that we will obtain.
It is probable, but not certain, that it will be near x0,
where the amplitude of the wavefunction is large.
After we perform the measurement, obtaining some result x, the wavefunction collapses into a position eigenstate
centered at x.
Wave functions can change as time progresses. An equation
known as the Schrödinger equation describes how wave
functions change in time, a role similar to
The Schrödinger equation, applied
to our free particle, predicts that the center of a wave packet will move
through space at a constant velocity, like a classical particle with no forces
acting on it. However, the wave packet will also spread out as time progresses,
which means that the position becomes more uncertain. This also has the effect
of turning position eigenstates (which can be thought
of as infinitely sharp wave packets) into broadened wave packets that are no
longer position eigenstates.
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,
a single electron in an unexcited atom is pictured classically as a particle
moving in a circular trajectory around the atomic nucleus, whereas in quantum
mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus.
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 not deterministic, but
rather unpredictable, i.e., random.
The probabilistic nature of quantum mechanics thus stems
from the act of measurement. This is one of the most difficult aspects of
quantum systems to understand. It was the central topic in the famous
Bohr-Einstein debates, in which the two scientists attempted to clarify these
fundamental principles by way of thought experiments.
In the decades after the formulation of quantum mechanics,
the question of what constitutes a "measurement" has been extensively
studied. Interpretations of quantum mechanics have been formulated to do away
with the concept of "wavefunction
collapse"; see, for example, the relative state interpretation.
The basic idea is that when a quantum system interacts with
a measuring apparatus, their respective wavefunctions
become entangled, so that the original quantum system ceases to exist as an
independent entity. For details, see the article on measurement in quantum
mechanics.
Quantum Mechanical Effects
As mentioned in the introduction, there are several classes
of phenomena that appear under quantum mechanics which have no analogue in
classical physics. These are sometimes referred to as "quantum
effects".
The first type of quantum effect is the quantization of
certain physical quantities. Quantization first arose in the mathematical
formulae of Max Planck in 1900 as discussed in the introduction. Max Planck was
analyzing how the radiation emitted from a body was related to its temperature,
in other words, he was analyzing the energy of a wave. The energy of a wave could
not be infinite, so Planck used the property of the wave we designate as the
frequency to define energy.
Max Planck discovered a constant that when multiplied by
the frequency of any wave gives the energy of the wave. This constant is
referred to by the letter h in mathematical formulae. It is a cornerstone of
physics. By measuring the energy in a discrete non-continuous portion of the
wave, the wave took on the appearance of chunks or packets of energy. These
chunks of energy resembled particles. So energy is said to be quantized because
it only comes in discrete chunks instead of a continuous range of energies.
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. Particle in a Box
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. In our free particle example, it
turns out that it is impossible to find a wavefunction
that is an eigenstate of both position and momentum.
This implies that position and momentum 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, never
both at the same 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". If quantum mechanics
is correct, entangled particles can display remarkable and counter-intuitive
properties. For example, a measurement made on one particle can produce,
through the collapse of the total wavefunction, an
instantaneous effect on other particles with which it is entangled, even if
they are far apart. (This does not conflict with special relativity because
information cannot be transmitted in this way.)
Mathematical
formulation of quantum mechanics
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 (variously called the "state space" or the
"associated Hilbert space" of the system).
The exact nature of this 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 proton is just the product of two complex
planes.
Each observable is represented by a densely defined Hermitian (or self-adjoint)
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.
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 time
evolution.
The inner product between two state vectors is a complex
number known as a probability amplitude. During a
measurement, the probability that a system collapses from a given initial state
to a particular eigenstate is given by the square of
the absolute value of the probability amplitudes between the initial and final
states. The possible results of a measurement are the eigenvalues
of the operator - which explains the choice of Hermitian
operators, for which all the eigenvalues are real.
We can 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 Schrödinger equation acts on
the entire probability amplitude, not merely its absolute value. Whereas the
absolute value of the probability amplitude encodes information about
probabilities, its phase encodes information about the interference between
quantum states.
This gives rise to the wave-like behavior of quantum states.It turns out that analytic solutions of Schrödinger's equation are only available for a small
number of model Hamiltonians, of which the quantum harmonic oscillator, the
particle in a box, the hydrogen-molecular ion and the hydrogen atom are the
most important representatives.
Even the helium atom, which contains just one more electron
than hydrogen, defies all attempts at a fully analytic treatment. There exist
several techniques for generating approximate solutions. For instance, in the
method known as perturbation theory one uses the analytic results for a simple
quantum mechanical model to generate results for a more complicated model
related to the simple model by, for example, the addition of a weak potential
energy. A
nother method is the "semi-classical equation of
motion" approach, which applies to systems for which quantum mechanics
produces weak deviations from classical behavior. The deviations can be
calculated based on the classical motion. This approach is important for the
field of quantum chaos.
An alternative formulation of quantum mechanics is
Feynman's path integral formulation, in which a quantum-mechanical
amplitude is considered as a sum over histories between initial and final
states; this is the quantum-mechanical counterpart of action principles in
classical mechanics.
Interactions with other scientific theories
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 physics when a system becomes large. This "large
system" limit is known as the classical or correspondence limit. One can
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.
Applications of quantum theory
Quantum mechanics has had enormous success in explaining
many of the features of our world. The individual behavior of the subatomic
particles that make up all forms of matter - electrons, protons, neutrons, and
so forth - can often only be satisfactorily described using quantum mechanics.Quantum mechanics has strongly influenced string
theory, a candidate for a theory of everything (see Reductionism. It is also
related to statistical mechanics.
Quantum mechanics is important for understanding how
individual atoms combine covalently to form chemicals or molecules. The
application of quantum mechanics to chemistry is known as quantum chemistry
(relativistic) quantum mechanics can in principle mathematically describe most
of chemistry.
Quantum mechanics can provide quantitative insight into
ionic and covalent bonding processes by explicitly showing which molecules are
energetically favorable to which others, and by approximately how much. Most of
the calculations performed in computational chemistry rely on quantum
mechanics.
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. 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
exponentially faster than classical computers. Another active research topic is
quantum teleportation, which deals with techniques to transmit quantum states
over arbitrary distances.
Since its inception, the many counter-intuitive results of
quantum mechanics have provoked strong philosophical debate and many
interpretations. Even fundamental issues such as Max Born's
basic rules concerning probability amplitudes and probability distributions
took decades to be appreciated.
The
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 there
should be a local hidden variable theory underlying
quantum mechanics and consequently the present theory was incomplete. He
produced a series of objections to the theory, the most famous of which has
become known as the EPR paradox.
John 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 that the
question is still not quite settled.
The
This is not accomplished by introducing some new axiom to
quantum mechanics, but on the contrary by removing the axiom of the collapse of
the wave packet: All the possible consistent states of the measured system and
the measuring apparatus (including the observer) are present in a real physical
(not just formally mathematical, as in other interpretations) quantum
superposition. (Such a superposition of consistent state combinations of
different systems is called an entangled state.)
While the multiverse is
deterministic, we perceive non-deterministic behavior governed by
probabilities, because we can observe only the universe, i.e. the consistent
state contribution to the mentioned superposition, we inhabit.
However, according to the theory of quantum decoherence, the parallel universes will never be accessible
for us, making them physically meaningless. This inaccessiblity
can be understood as follows: once a measurement is done, the measured system
becomes entangled with both the physicist who measured it and a huge number of
other particles, some of which are photons flying away towards the other end of
the universe; in order to prove that the wave function did not collapse one
would have to bring all these particles back and measure them again, together
with the system that was measured originally. This is completely impractical,
but even if one can theoretically do this, it would destroy any evidence that
the original measurement took place (including the physicist's memory).
History
In 1900 the German physicist 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. The idea that
each photon had to consist of energy in terms of quanta was a remarkable
achievement as it effectively removed the possibility of black body radiation
attaining infinite energy if it were to be explained in terms of wave forms
only.
In 1913, Bohr explained the spectral lines of the hydrogen
atom, again by using quantization, in his paper of July 1913 'On the
Constitution of Atoms and Molecules'.
In 1924, the French physicist Louis de Broglie
put forward his theory of matter waves by stating that particles can exhibit
wave characteristics and vice versa.
These theories, though successful, were strictly
phenomenological: there was no rigorous justification for quantization (aside,
perhaps, for Henri Poincaré's discussion of Planck's
theory in his 1912 paper "Sur la théorie des quanta"). They are collectively known as
the old quantum theory.
The phrase "quantum physics" was first used in
Modern quantum mechanics was born in 1925, when the German
physicist Heisenberg developed matrix mechanics and the Austrian physicist Schrödinger invented wave mechanics and the
non-relativistic Schrödinger equation. Schrödinger subsequently showed that the two approaches
were equivalent.
Heisenberg formulated his uncertainty principle in 1927,
and the
The field of quantum chemistry was pioneered by physicists
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
theoretical chemist Linus Pauling
at Cal Tech, and John Slater into various theories such as Molecular Orbital
Theory or Valence Theory.
Beginning in 1927, attempts were made to apply quantum
mechanics to fields rather than single particles, resulting in what are known
as quantum field theories. Early workers in this area included Dirac, Pauli, Weisskopf,
and Jordan. This area of research culminated in the formulation of quantum
electrodynamics by Feynman, Dyson, Schwinger, and Tomonaga during the 1940s.
Quantum electrodynamics is a quantum theory of electrons,
positrons, and the electromagnetic field, and served as a role model for
subsequent quantum field theories.
The theory of quantum chromodynamics
was formulated beginning in the early 1960s. The theory as we know it today was
formulated by Politzer, Gross and Wilzcek
in 1975.
Building on pioneering work by Schwinger,
Higgs, Goldstone, Glashow, Weinberg and Salam independently showed how the weak nuclear force and
quantum electrodynamics could be merged into a single electroweak force.