A wave function or wavefunction is a mathematical tool used in quantum mechanics Quantum mechanics , also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic scales. In advanced topics of QM, some of these. It is a function The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain typically of space or momentum or spin and possibly of time that returns the probability amplitude In quantum mechanics, a probability amplitude is a complex number whose absolute value squared represents a probability or probability density. For example, the values taken by a normalised wave function ψ are amplitudes, since |ψ|2 gives the probability density at position x. Probability amplitudes may also correspond to probabilities of of a position or momentum for a subatomic particle In physics, subatomic particles are the small particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles. Particle physics and nuclear physics study these particles and how they interact. Mathematically, it is a function from a space that maps the possible states of the system into the complex numbers A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the. The laws of quantum mechanics (the Schrödinger equation In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics) describe how the wave function evolves over time.

The electron probability density for the first few hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force. The most abundant isotope, hydrogen-1, protium, or light hydrogen, contains no neutrons; other isotopes of hydrogen, such as electron orbitals An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. These functions may serve as three-dimensional graph of an electron shown as cross-sections. These orbitals form an orthonormal basis In mathematics, an orthonormal basis of an inner product space V , is a set of mutually orthogonal vectors of magnitude 1 (unit vectors) that span the space when infinite linear combinations are allowed. (In some contexts, especially in linear algebra, the concept of basis means a set of vectors that span a space when only finite linear for the wave function of the electron. Different orbitals are depicted with different scale.
Quantum mechanics Quantum mechanics , also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic scales. In advanced topics of QM, some of these
Uncertainty principle In quantum mechanics, the Heisenberg uncertainty principle states by precise inequalities that certain pairs of physical properties, such as position and momentum, cannot be simultaneously known to arbitrarily high precision. That is, the more precisely one property is measured, the less precisely the other can be measured. The principle states
Introduction Everyday experience creates preconceptions that fail drastically when that experience is extended to the very massive and the very fast, or when extended to the very small and the very cold. The large scale requires relativity theory, and the small scale requires quantum mechanics. Quantum physics deals with "Nature as She is—absurd." · Mathematical formulations
Background
Classical mechanics In the fields of physics, classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain boundary under the action of a system of forces Old quantum theory The old quantum theory was a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics. The Bohr model was the focus of study, and Interference In physics, interference is the addition of two or more waves that results in a new wave pattern. Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference in physics corresponds to · Bra-ket notation Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in pure mathematics. It is so called because the inner product of two states is denoted by a bracket, , consisting of a left Hamiltonian In quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system. It is of fundamental importance in most formulations of quantum theory because of its close relation to the time-evolution of a system
Fundamental concepts
Quantum state In quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be statistically mixed, corresponding to an · Wave function Superposition The principle of superposition states that if the world can be in any configuration, any possible arrangement of particles or fields, and if the world could also be in another configuration, then the world can also be in a state which is a superposition of the two, where the amount of each configuration that is in the superposition is specified by · Entanglement Quantum entanglement, also called the quantum non-local connection, is a property of a quantum mechanical state of a system of two or more objects in which the quantum states of the constituting objects are linked together so that one object can no longer be adequately described without full mention of its counterpart—even if the individual Complementarity · Duality · Uncertainty In quantum mechanics, the Heisenberg uncertainty principle states by precise inequalities that certain pairs of physical properties, such as position and momentum, cannot be simultaneously known to arbitrarily high precision. That is, the more precisely one property is measured, the less precisely the other can be measured. The principle states Measurement The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus · Exclusion The Pauli exclusion principle is a quantum mechanical principle formulated by the Austrian physicist Wolfgang Pauli in 1925. In its simplest form for electrons in a single atom, it states that no two electrons can have the same four quantum numbers, that is, if n, l, and ml are the same, ms must be different such that the electrons have opposite Decoherence In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior. Quantum decoherence gives the appearance of wave function collapse and justifies the framework and intuition of classical physics as an acceptable approximation: decoherence is the · Ehrenfest theorem The Ehrenfest theorem, named after Paul Ehrenfest, the Austrian physicist and mathematician, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. It is · Tunnelling Quantum tunneling refers to the phenomena of a particle's ability to penetrate energy barriers within electronic structures. The scientific terms for this are Wave-mechanical tunneling, Quantum-mechanical tunneling and the Tunnel effect. The Tunnel Effect is an evanescent wave coupling effect that occurs in the context of quantum mechanics
Experiments
Double-slit experiment In quantum mechanics, the double-slit experiment demonstrates the inseparability of the wave and particle natures of light and other quantum particles. A coherent light source (e.g., a laser) illuminates a thin plate with two parallel slits cut in it, and the light passing through the slits strikes a screen behind them. The wave nature of light Davisson–Germer experiment Stern–Gerlach experiment Bell's inequality experiment The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality. John Bell published the first inequality of this kind in his paper "On the Einstein-Podolsky-Rosen Paradox". Bell's Theorem states that a Bell inequality must be obeyed under any local hidden Popper's experiment Popper's experiment is an experiment proposed by the 20th century philosopher of science Karl Popper, an advocate of strict scientific method who opposed the Copenhagen interpretation, to test that standard interpretation of quantum mechanics. Popper's experiment is similar in spirit to the thought experiment of Einstein, Podolsky and Rosen Schrödinger's cat Schrödinger's cat is a thought experiment, often described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The thought experiment presents a cat that might be alive or dead, depending on an earlier Elitzur–Vaidman bomb-tester Quantum eraser In physics, the quantum eraser experiment is a double-slit experiment that demonstrates several laws of quantum mechanics, including wave-particle duality, which seeks to explain certain wave and particle properties of matter, complementarity, and the Copenhagen interpretation of quantum mechanics, which outlines the idea that, in quantum
Formulations
Schrödinger picture Heisenberg picture Interaction picture Matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 Sum over histories The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude
Equations
Schrödinger equation In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics Pauli equation The Pauli Equation, also known as the Schrödinger–Pauli equation, is the formulation of the Schrödinger equation for spin-1⁄2 particles which takes into account the interaction of the particle's spin with the electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are slow enough that Klein–Gordon equation Dirac equation In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 which provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. The equation demands the existence of Rydberg formula The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. The formula was invented by the Swedish physicist Johannes Rydberg and presented on November 5, 1888
Interpretations An interpretation of quantum mechanics is a statement which attempts to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has received thorough experimental testing, many of these experiments are open to different interpretations. There exist a number of contending schools of thought, differing over
de Broglie–Bohm · CCC · Consistent histories In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. This interpretation of quantum mechanics is based on a consistency criterion that then allows probabilities to · Copenhagen The Copenhagen interpretation is an interpretation of quantum mechanics. A key feature of quantum mechanics is that the state of every particle is described by a wavefunction, which is a mathematical representation used to calculate the probability for it to be found in a location or a state of motion. According to this interpretation, the act of · Ensemble The Ensemble Interpretation, or Statistical Interpretation of quantum mechanics, is an interpretation that can be viewed as a minimalist interpretation; it is a quantum mechanical interpretation that claims to make the fewest assumptions associated with the standard mathematical formalization. At its heart, it takes the statistical interpretation · Hidden variables Historically, in physics, hidden variable theories were espoused by a minority of physicists who argued that the statistical nature of quantum mechanics indicated that quantum mechanics is "incomplete". Albert Einstein, the most famous proponent of hidden variables, insisted that, "I am convinced God does not play dice" — · Many-worlds Many-worlds is an interpretation of quantum mechanics that asserts the objective reality of the wavefunction, but denies the reality of wavefunction collapse. It is also known as MWI, the relative state formulation, theory of the universal wavefunction, parallel universes, many-universes interpretation or just many worlds · Pondicherry · Quantum logic In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account. This research area and its name originated in the 1936 paper by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical boolean logic with the · Relational Relational quantum mechanics is an interpretation of quantum mechanics which treats the state of a quantum system as being observer-dependent, that is, the state is the relation between the observer and the system. This interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and has since been expanded upon by a number of · Stochastic · Transactional The transactional interpretation of quantum mechanics describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves. It was first proposed in 1986 by John G. Cramer, who argues that it helps in developing intuition for quantum processes, avoids the philosophical problems with · Objective collapse
Advanced topics
Quantum information science Scattering theory In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow. Scattering also includes the interaction of Quantum field theory Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized (represented) by an infinite number of dynamical degrees of freedom, that is, fields and (in a condensed matter context) many-body systems. It is the natural and quantitative language of particle physics and Quantum chaos Quantum chaos is a branch of physics which studies how chaotic classical systems can be shown to be limits of quantum-mechanical systems. The phenomena covered by quantum chaos so far are mainly related to wave theory. An alternative name, proposed by Sir Michael Berry, is quantum chaology. Michael Berry argues that there is no quantum chaos, in
Scientists
Bell John Stewart Bell was a physicist, and the originator of Bell's Theorem, one of the most important theorems in quantum physics • Bohm • Bohr • Born • Bose • de Broglie • Dirac • Ehrenfest • Everett • Feynman • Heisenberg • Jordan • Kramers • von Neumann • Pauli • Planck • Salam• Schrödinger • Sommerfeld • Wien • Wigner

It is commonly applied as a property of particles relating to their wave-particle duality, where it is denoted ψ(position,time) and where | ψ | 2 is equal to the chance of finding the subject at a certain time and position.[1] For example, in an atom with a single electron, such as hydrogen or ionized helium, the wave function of the electron provides a complete description of how the electron behaves. It can be decomposed into a series of atomic orbitals which form a basis for the possible wave functions. For atoms with more than one electron (or any system with multiple particles), the underlying space is the possible configurations of all the electrons and the wave function describes the probabilities of those configurations.

A simple wave function is that for a particle in a box. Another simple example is a free particle (or a particle in a large box), whose wave function is a sinusoidal where, in the spirit of the uncertainty principle, the momentum is known but the position is not known.

Contents

Definition

The modern usage of the term wave function refers to a complex vector or function, i.e. an element in a complex Hilbert space. Typically, a wave function is either:

,
,
.

In all cases, the wave function provides a complete description of the associated physical system. An element of a vector space can be expressed in different bases; and so the same applies to wave functions. The components of a wave function describing the same physical state take different complex values depending on the basis being used; however the wave function itself is not dependent on the basis chosen. In this respect they are like spatial vectors in ordinary space because choosing a new set of cartesian axes by rotation of the coordinate frame does not alter the vector itself, only the representation of the vector with respect to the coordinate frame. A basis in quantum mechanics is analogous to the coordinate frame in that choosing a new basis does not alter the wavefunction, only its representation, which is expressed as the values of the components above.

Because the probabilities that the system is in each possible state should add up to 1, the norm of the wave function must be 1.

Spatial interpretation

The physical interpretation of the wave function is context dependent. Several examples are provided below, followed by a detailed discussion of the three cases described above.

One particle in one spatial dimension

The spatial wave function associated with a particle in one dimension is a complex function defined over the real line. The positive function is interpreted as the probability density associated with the particle's position. That is, the probability of a measurement of the particle's position yielding a value in the interval [a,b] is given by

.

This leads to the normalization condition

.

since the probability of a measurement of the particle's position yielding a value in the range is unity.

One particle in three spatial dimensions

3D confined electron wave functions in a Quantum Dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ‘s-type’ and ‘p-type’. However, in a triangular dot the wave functions are mixed due to confinement symmetry.

The three dimensional case is analogous to the one dimensional case; the wave function is a complex function defined over three dimensional space, and the square of its absolute value is interpreted as a three dimensional probability density function:

The normalization condition is likewise

where the preceding integral is taken over all space.

Two distinguishable particles in three spatial dimensions

In this case, the wave function is a complex function of six spatial variables, , and is the joint probability density associated with the positions of both particles. Thus the probability that a measurement of the positions of both particles indicates particle one is in region R and particle two is in region S is

where dV1 = dx1dy1dz1, and similarly for dV2.

The normalization condition is then:

in which the preceding integral is taken over the full range of all six variables.

Given a wave function ψ of a system consisting of two (or more) particles, it is in general not possible to assign a definite wave function to a single-particle subsystem. In other words, the particles in the system can be entangled.

Show All>>

 

The above information uses material from Wikipedia and is licensed under the GNU Free Documentation License.
Some facts may not have been fully verified for accuracy. [Disclaimers]
This page was last archived by our server on Mon Sep 6 04:38:12 2010. [ refresh local cache ]
Displaying this page or its contents does not use any Wikimedia Foundation's resources.
The owners of this site proudly support the Wikimedia Foundation.

[Hide]▼
[Hide]▲