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A wave function in
quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
is a mathematical description of the quantum state of an isolated
quantum system Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
. The wave function is a
complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi, respectively). The wave function is a function of the
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
corresponding to some maximal set of commuting
observables In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
. Some particles, like electrons and photons, have nonzero
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, and the wave function for such particles include spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin) – these values are often displayed in a
column matrix In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, ...
(e.g., a column vector for a non-relativistic electron with spin ). According to the
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The inner product between two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves. In
Born Born may refer to: * Childbirth * Born (surname), a surname (see also for a list of people with the name) * ''Born'' (comics), a comic book limited series Places * Born, Belgium, a village in the German-speaking Community of Belgium * Born, Luxe ...
's statistical interpretation in non-relativistic quantum mechanics, the
squared modulus In mathematics, a square is the result of multiplication, multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as exponentiation, raising to the power 2 (number), 2, and is denoted by a ...
of the wave function, , is a real number interpreted as the probability density of measuring a particle as being at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the ''normalization condition''. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured—its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function and calculate the statistical distributions for measurable quantities.


Historical background

In 1905, Albert Einstein postulated the proportionality between the frequency f of a photon and its energy and in 1916 the corresponding relation between a photon's
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
p and wavelength where h is the Planck constant. In 1923, De Broglie was the first to suggest that the relation now called the De Broglie relation, holds for ''massive'' particles, the chief clue being Lorentz invariance, and this can be viewed as the starting point for the modern development of quantum mechanics. The equations represent wave–particle duality for both massless and massive particles. In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing " wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
, and others, developing " matrix mechanics". Schrödinger subsequently showed that the two approaches were equivalent. In 1926, Schrödinger published the famous wave equation now named after him, the Schrödinger equation. This equation was based on classical
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
using quantum operators and the de Broglie relations and the solutions of the equation are the wave functions for the quantum system. However, no one was clear on how to interpret it. At first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large. This was shown to be incompatible with the elastic scattering of a wave packet (representing a particle) off a target; it spreads out in all directions., translated in at pages 52–55. While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of probability amplitude., translated in . Als
here
This relates calculations of quantum mechanics directly to probabilistic experimental observations. It is accepted as part of the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics. In 1927, Hartree and Fock made the first step in an attempt to solve the ''N''-body wave function, and developed the ''self-consistency cycle'': an iterative algorithm to approximate the solution. Now it is also known as the Hartree–Fock method. The Slater determinant and permanent (of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
) was part of the method, provided by John C. Slater. Schrödinger did encounter an equation for the wave function that satisfied relativistic energy conservation ''before'' he published the non-relativistic one, but discarded it as it predicted negative
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
and negative energies. In 1927, Klein, Gordon and Fock also found it, but incorporated the electromagnetic interaction and proved that it was Lorentz invariant. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the Klein–Gordon equation. In 1927,
Pauli Pauli is a surname and also a Finnish male given name (variant of Paul) and may refer to: * Arthur Pauli (born 1989), Austrian ski jumper * Barbara Pauli (1752 or 1753 - fl. 1781), Swedish fashion trader *Gabriele Pauli (born 1957), German politi ...
phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the Pauli equation. Pauli found the wave function was not described by a single complex function of space and time, but needed two complex numbers, which respectively correspond to the spin +1/2 and −1/2 states of the fermion. Soon after in 1928, Dirac found an equation from the first successful unification of special relativity and quantum mechanics applied to the electron, now called the Dirac equation. In this, the wave function is a ''spinor'' represented by four complex-valued components: two for the electron and two for the electron's antiparticle, the
positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...
. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, other relativistic wave equations were found.


Wave functions and wave equations in modern theories

All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic counterparts. The Klein–Gordon equation and the Dirac equation, while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called relativistic quantum mechanics, while very successful, has its limitations (see e.g. Lamb shift) and conceptual problems (see e.g. Dirac sea). Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation,
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
is needed. In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called ''field operators'' (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the ''free fields operators'', i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases. Thus the Klein–Gordon equation (spin ) and the Dirac equation (spin ) in this guise remain in the theory. Higher spin analogues include the
Proca equation In physics, specifically field theory (physics), field theory and particle physics, the Proca action describes a massive spin (physics), spin-1 quantum field, field of mass ''m'' in Minkowski spacetime. The corresponding equation is a relativisti ...
(spin ), Rarita–Schwinger equation (spin ), and, more generally, the
Bargmann–Wigner equations :''This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators. In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non ...
. For ''massless'' free fields two examples are the free field
Maxwell equation Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
(spin ) and the free field Einstein equation (spin ) for the field operators. All of them are essentially a direct consequence of the requirement of Lorentz invariance. Their solutions must transform under Lorentz transformation in a prescribed way, i.e. under a particular
representation of the Lorentz group The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of repres ...
and that together with few other reasonable demands, e.g. the cluster decomposition property, with implications for
causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
is enough to fix the equations. This applies to free field equations; interactions are not included. If a Lagrangian density (including interactions) is available, then the Lagrangian formalism will yield an equation of motion at the classical level. This equation may be very complex and not amenable to solution. Any solution would refer to a ''fixed'' number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory. In
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
, the situation remains analogous. For instance, a wave function in momentum space has the role of Fourier expansion coefficient in a general state of a particle (string) with momentum that is not sharply defined.


Definition (one spinless particle in one dimension)

For now, consider the simple case of a non-relativistic single particle, without
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, in one spatial dimension. More general cases are discussed below.


Position-space wave functions

The state of such a particle is completely described by its wave function, \Psi(x,t)\,, where is position and is time. This is a complex-valued function of two real variables and . For one spinless particle in one dimension, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number \left, \Psi(x, t)\^2 = \Psi^*(x, t)\Psi(x, t) = \rho(x, t), is interpreted as the probability density that the particle is at . The asterisk indicates the complex conjugate. If the particle's position is
measured Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared t ...
, its location cannot be determined from the wave function, but is described by a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.


Normalization condition

The probability that its position will be in the interval is the integral of the density over this interval: P_ (t) = \int_a^b \,, \Psi(x,t), ^2 dx where is the time at which the particle was measured. This leads to the normalization condition: \int_^\infty \, , \Psi(x,t), ^2dx = 1\,, because if the particle is measured, there is 100% probability that it will be ''somewhere''. For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). Technically, because of the normalization condition, wave functions form a
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
rather than an ordinary vector space. This vector space is infinite- dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, because the inner product of two wave functions and can be defined as the complex number (at time )The functions are here assumed to be elements of , the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. This is necessary to obtain an inner product (that is, ) as opposed to a semi-inner product. The integral is taken to be the Lebesgue integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space.
( \Psi_1 , \Psi_2 ) = \int_^\infty \, \Psi_1^*(x, t)\Psi_2(x, t)dx. More details are given
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
. Although the inner product of two wave functions is a complex number, the inner product of a wave function with itself, (\Psi,\Psi) = \, \Psi\, ^2 \,, is ''always'' a positive real number. The number (not ) is called the norm of the wave function . If , then is normalized. If is not normalized, then dividing by its norm gives the normalized function . Two wave functions and are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
if . If they are normalized ''and'' orthogonal, they are orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions. In a linear combination of orthogonal wave functions we have, \Psi = \sum_n a_n \Psi_n \,,\quad a_n = \frac If the wave functions were nonorthogonal, the coefficients would be less simple to obtain.


Quantum states as vectors

In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number \left, (\Psi_1,\Psi_2)\^2 = P\left(\Psi_2 \rightarrow \Psi_1\right) \,, which, assuming both wave functions are normalized, is interpreted as the probability of the wave function "collapsing" to the new wave function upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with being an eigenvector of the resulting eigenvalue. This is the Born rule, and is one of the fundamental postulates of quantum mechanics. At a particular instant of time, all values of the wave function are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In
Bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
, this vector is written , \Psi(t)\rangle = \int\Psi(x,t) , x\rangle dx and is referred to as a "quantum state vector", or simply "quantum state". There are several advantages to understanding wave functions as representing elements of an abstract vector space: * All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example: ** Linear algebra explains how a vector space can be given a basis, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space and suggests that there are other possibilities too. **
Bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
can be used to manipulate wave functions. * The idea that quantum states are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations. The time parameter is often suppressed, and will be in the following. The coordinate is a continuous index. The are the basis vectors, which are orthonormal so their inner product is a delta function; \langle x' , x \rangle = \delta(x' - x) thus \langle x' , \Psi\rangle = \int \Psi(x) \langle x', x\rangle dx= \Psi(x') and , \Psi\rangle = \int , x\rangle \langle x , \Psi\rangle dx= \left( \int , x\rangle \langle x , dx\right) , \Psi\rangle which illuminates the identity operator I = \int , x\rangle \langle x , dx\,. Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis).


Momentum-space wave functions

The particle also has a wave function in momentum space: \Phi(p,t) where is the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
in one dimension, which can be any value from to , and is time. Analogous to the position case, the inner product of two wave functions and can be defined as: (\Phi_1 , \Phi_2 ) = \int_^\infty \, \Phi_1^*(p, t)\Phi_2(p, t) dp\,. One particular solution to the time-independent Schrödinger equation is \Psi_p(x) = e^, a plane wave, which can be used in the description of a particle with momentum exactly , since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they aren't square-integrable), so they are not really elements of physical Hilbert space. The set \ forms what is called the momentum basis. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions aren't normalizable, they are instead normalized to a delta function,Also called "Dirac orthonormality", according to (\Psi_,\Psi_) = \delta(p - p'). For another thing, though they are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to express all functions in it using Fourier transforms as described next.


Relations between position and momentum representations

The and representations are \begin , \Psi\rangle = I, \Psi\rangle &= \int , x\rangle \langle x, \Psi\rangle dx = \int \Psi(x) , x\rangle dx,\\ , \Psi\rangle = I, \Psi\rangle &= \int , p\rangle \langle p, \Psi\rangle dp = \int \Phi(p) , p\rangle dp. \end Now take the projection of the state onto eigenfunctions of momentum using the last expression in the two equations, \int \Psi(x) \langle p, x\rangle dx = \int \Phi(p') \langle p, p'\rangle dp' = \int \Phi(p') \delta(p-p') dp' = \Phi(p). Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation \langle x , p \rangle = p(x) = \frace^ \Rightarrow \langle p , x \rangle = \frace^, one obtains \Phi(p) = \frac\int \Psi(x)e^dx\,. Likewise, using eigenfunctions of position, \Psi(x) = \frac\int \Phi(p)e^dp\,. The position-space and momentum-space wave functions are thus found to be
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
s of each other. The two wave functions contain the same information, and either one alone is sufficient to calculate any property of the particle. As representatives of elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same state vector, hence ''identical physical states'', but they are not generally equal when viewed as square-integrable functions. In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
, and enter symmetrically, so there it doesn't matter which description one uses. The same equation (modulo constants) results. From this follows, with a little bit of afterthought, a factoid: The solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in .The Fourier transform viewed as a unitary operator on the space has eigenvalues . The eigenvectors are "Hermite functions", i.e.
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
multiplied by a Gaussian function. See for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.


Definitions (other cases)

Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.


One-particle states in 3d position space

The position-space wave function of a single particle without spin in three spatial dimensions is similar to the case of one spatial dimension above: \Psi(\mathbf,t) where is the
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
in three-dimensional space, and is time. As always is a complex-valued function of real variables. As a single vector in Dirac notation , \Psi(t)\rangle = \int d^3\! \mathbf\, \Psi(\mathbf,t) \,, \mathbf\rangle All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions. For a particle with
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter); \xi(s_z,t) where is the spin projection quantum number along the axis. (The axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The parameter, unlike and , is a ''discrete variable''. For example, for a spin-1/2 particle, can only be or , and not any other value. (In general, for spin , can be ). Inserting each quantum number gives a complex valued function of space and time, there are of them. These can be arranged into a column vectorColumn vectors can be motivated by the convenience of expressing the spin operator for a given spin as a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, for the z-component spin operator (divided by hbar to nondimensionalize): \frac\hat_z = \begin s & 0 & \cdots & 0 & 0 \\ 0 & s-1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -(s-1) & 0 \\ 0 & 0 & \cdots & 0 & -s \end The eigenvectors of this matrix are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.
\xi = \begin \xi(s,t) \\ \xi(s-1,t) \\ \vdots \\ \xi(-(s-1),t) \\ \xi(-s,t) \\ \end = \xi(s,t) \begin 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end + \xi(s-1,t)\begin 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end + \cdots + \xi(-(s-1),t) \begin 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end + \xi(-s,t) \begin 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end In
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
, these easily arrange into the components of a vectorEach is usually identified as a column vector: , s\rangle \leftrightarrow \begin 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end \,, \quad , s-1\rangle \leftrightarrow \begin 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end \,, \ldots \,, \quad , -(s-1)\rangle \leftrightarrow \begin 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end \,,\quad , -s\rangle \leftrightarrow \begin 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end but it is a common abuse of notation to write: , s\rangle = \begin 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end \, \ldots \,, because the kets are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components. , \xi (t)\rangle = \sum_^s \xi(s_z,t) \,, s_z \rangle The entire vector is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of ordinary differential equations with solutions . The term "spin function" instead of "wave function" is used by some authors. This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation. More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as: \Psi(\mathbf,s_z,t) and these can also be arranged into a column vector \Psi(\mathbf,t) = \begin \Psi(\mathbf,s,t) \\ \Psi(\mathbf,s-1,t) \\ \vdots \\ \Psi(\mathbf,-(s-1),t) \\ \Psi(\mathbf,-s,t) \\ \end in which the spin dependence is placed in indexing the entries, and the wave function is a complex vector-valued function of space and time only. All values of the wave function, not only for discrete but continuous variables also, collect into a single vector , \Psi(t)\rangle = \sum_\int d^3\!\mathbf \,\Psi(\mathbf,s_z,t)\, , \mathbf, s_z\rangle For a single particle, the tensor product of its position state vector and spin state vector gives the composite position-spin state vector , \psi(t)\rangle\! \otimes\! , \xi(t)\rangle = \sum_\int d^3\! \mathbf\, \psi(\mathbf,t)\,\xi(s_z,t) \,, \mathbf\rangle \!\otimes\! , s_z\rangle with the identifications , \Psi (t)\rangle = , \psi(t)\rangle \!\otimes\! , \xi(t)\rangle \Psi(\mathbf,s_z,t) = \psi(\mathbf,t)\,\xi(s_z,t) , \mathbf,s_z \rangle= , \mathbf\rangle \!\otimes\! , s_z\rangle The tensor product factorization is only possible if the orbital and spin angular momenta of the particle are separable in the Hamiltonian operator underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms). The time dependence can be placed in either factor, and time evolution of each can be studied separately. The factorization is not possible for those interactions where an external field or any space-dependent quantity couples to the spin; examples include a particle in a
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, and spin–orbit coupling. The preceding discussion is not limited to spin as a discrete variable, the total angular momentum ''J'' may also be used. Other discrete degrees of freedom, like isospin, can expressed similarly to the case of spin above.


Many-particle states in 3d position space

If there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that ''one'' wave function describes ''many'' particles is what makes quantum entanglement and the EPR paradox possible. The position-space wave function for particles is written: \Psi(\mathbf_1,\mathbf_2 \cdots \mathbf_N,t) where is the position of the -th particle in three-dimensional space, and is time. Altogether, this is a complex-valued function of real variables. In quantum mechanics there is a fundamental distinction between '' identical particles'' and ''distinguishable'' particles. For example, any two electrons are identical and fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it. This translates to a requirement on the wave function for a system of identical particles: \Psi \left ( \ldots \mathbf_a, \ldots , \mathbf_b, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf_b, \ldots , \mathbf_a, \ldots \right ) where the sign occurs if the particles are ''all bosons'' and sign if they are ''all fermions''. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions. The physical interchange of particles corresponds to mathematically switching arguments in the wave function. The antisymmetry feature of fermionic wave functions leads to the Pauli principle. Generally, bosonic and fermionic symmetry requirements are the manifestation of particle statistics and are present in other quantum state formalisms. For ''distinguishable'' particles (no two being identical, i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric. For a collection of particles, some identical with coordinates and others distinguishable (not identical with each other, and not identical to the aforementioned identical particles), the wave function is symmetric or antisymmetric in the identical particle coordinates only: \Psi \left ( \ldots \mathbf_a, \ldots , \mathbf_b, \ldots , \mathbf_1, \mathbf_2, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf_b, \ldots , \mathbf_a, \ldots , \mathbf_1, \mathbf_2, \ldots \right ) Again, there is no symmetry requirement for the distinguishable particle coordinates . The wave function for ''N'' particles each with spin is the complex-valued function \Psi(\mathbf_1, \mathbf_2 \cdots \mathbf_N, s_, s_ \cdots s_, t) Accumulating all these components into a single vector, , \Psi \rangle = \overbrace^ \overbrace^ \; \underbrace_ \; \underbrace_\,. For identical particles, symmetry requirements apply to both position and spin arguments of the wave function so it has the overall correct symmetry. The formulae for the inner products are integrals over all coordinates or momenta and sums over all spin quantum numbers. For the general case of particles with spin in 3-d, ( \Psi_1 , \Psi_2 ) = \sum_ \cdots \sum_ \sum_ \int\limits_ d ^3\mathbf_1 \int\limits_ d ^3\mathbf_2\cdots \int\limits_ d ^3 \mathbf_N \Psi^_1 \left(\mathbf_1 \cdots \mathbf_N,s_\cdots s_,t \right )\Psi_2 \left(\mathbf_1 \cdots \mathbf_N,s_\cdots s_,t \right ) this is altogether three-dimensional volume integrals and sums over the spins. The differential volume elements are also written "" or "". The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.


Probability interpretation

For the general case of particles with spin in 3d, if is interpreted as a probability amplitude, the probability density is \rho\left(\mathbf_1 \cdots \mathbf_N,s_\cdots s_,t \right ) = \left , \Psi\left (\mathbf_1 \cdots \mathbf_N,s_\cdots s_,t \right ) \right , ^2 and the probability that particle 1 is in region with spin ''and'' particle 2 is in region with spin etc. at time is the integral of the probability density over these regions and evaluated at these spin numbers: P_ (t) = \int_ d ^3\mathbf_1 \int_ d ^3\mathbf_2\cdots \int_ d ^3\mathbf_N \left , \Psi\left (\mathbf_1 \cdots \mathbf_N,m_1\cdots m_N,t \right ) \right , ^2


Time dependence

For systems in time-independent potentials, the wave function can always be written as a function of the degrees of freedom multiplied by a time-dependent phase factor, the form of which is given by the Schrödinger equation. For particles, considering their positions only and suppressing other degrees of freedom, \Psi(\mathbf_1,\mathbf_2,\ldots,\mathbf_N,t) = e^ \,\psi(\mathbf_1,\mathbf_2,\ldots,\mathbf_N)\,, where is the energy eigenvalue of the system corresponding to the eigenstate . Wave functions of this form are called stationary states. The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states. For any quantum state and operator , in the Schrödinger picture changes with time according to the Schrödinger equation while is constant. In the Heisenberg picture it is the other way round, is constant while evolves with time according to the Heisenberg equation of motion. The Dirac (or interaction) picture is intermediate, time dependence is places in both operators and states which evolve according to equations of motion. It is useful primarily in computing S-matrix elements.


Non-relativistic examples

The following are solutions to the Schrödinger equation for one nonrelativistic spinless particle.


Finite potential barrier

One of most prominent features of the wave mechanics is a possibility for a particle to reach a location with a prohibitive (in classical mechanics) force potential. A common model is the " potential barrier", the one-dimensional case has the potential V(x)=\beginV_0 & , x, and the steady-state solutions to the wave equation have the form (for some constants ) \Psi (x) = \begin A_e^+A_e^ & x<-a, \\ B_e^+B_e^ & , x, \le a, \\ C_e^+C_e^ & x>a. \end Note that these wave functions are not normalized; see
scattering theory In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunli ...
for discussion. The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative ): setting corresponds to firing particles singly; the terms containing and signify motion to the right, while and – to the left. Under this beam interpretation, put since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above. In a semiconductor crystallite whose radius is smaller than the size of its
exciton An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. The ...
Bohr radius, the excitons are squeezed, leading to quantum confinement. The energy levels can then be modeled using the particle in a box model in which the energy of different states is dependent on the length of the box.


Quantum harmonic oscillator

The wave functions for the
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...
can be expressed in terms of Hermite polynomials , they are \Psi_n(x) = \sqrt \cdot \left(\frac\right)^ \cdot e^ \cdot H_n\left(\sqrt x \right) where .


Hydrogen atom

The wave functions of an electron in a
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. Atomic hydrogen consti ...
are expressed in terms of spherical harmonics and generalized Laguerre polynomials (these are defined differently by different authors—see main article on them and the hydrogen atom). It is convenient to use spherical coordinates, and the wave function can be separated into functions of each coordinate,Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, \Psi_(r,\theta,\phi) = R(r)\,\,Y_\ell^m\!(\theta, \phi) where are radial functions and are spherical harmonics of degree and order . This is the only atom for which the Schrödinger equation has been solved exactly. Multi-electron atoms require approximative methods. The family of solutions is: \Psi_(r,\theta,\phi) = \sqrt e^ \left(\frac\right)^ L_^\left(\frac\right) \cdot Y_^(\theta, \phi ) where is the Bohr radius, are the generalized Laguerre polynomials of degree , is the principal quantum number, the azimuthal quantum number, the magnetic quantum number. Hydrogen-like atoms have very similar solutions. This solution does not take into account the spin of the electron. In the figure of the hydrogen orbitals, the 19 sub-images are images of wave functions in position space (their norm squared). The wave functions represent the abstract state characterized by the triple of quantum numbers , in the lower right of each image. These are the principal quantum number, the orbital angular momentum quantum number, and the magnetic quantum number. Together with one spin-projection quantum number of the electron, this is a complete set of observables. The figure can serve to illustrate some further properties of the function spaces of wave functions. * In this case, the wave functions are square integrable. One can initially take the function space as the space of square integrable functions, usually denoted . * The displayed functions are solutions to the Schrödinger equation. Obviously, not every function in satisfies the Schrödinger equation for the hydrogen atom. The function space is thus a subspace of . * The displayed functions form part of a basis for the function space. To each triple , there corresponds a basis wave function. If spin is taken into account, there are two basis functions for each triple. The function space thus has a
countable basis In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel basis, Hamel) basis (linear algebra), basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bas ...
. * The basis functions are mutually orthonormal.


Wave functions and function spaces

The concept of
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s enters naturally in the discussion about wave functions. A function space is a set of functions, usually with some defining requirements on the functions (in the present case that they are square integrable), sometimes with an
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
on the set (in the present case a vector space structure with an inner product), together with a topology on the set. The latter will sparsely be used here, it is only needed to obtain a precise definition of what it means for a subset of a function space to be
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
. It will be concluded below that the function space of wave functions is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. This observation is the foundation of the predominant mathematical formulation of quantum mechanics.


Vector space structure

A wave function is an element of a function space partly characterized by the following concrete and abstract descriptions. * The Schrödinger equation is linear. This means that the solutions to it, wave functions, can be added and multiplied by scalars to form a new solution. The set of solutions to the Schrödinger equation is a vector space. * The superposition principle of quantum mechanics. If and are two states in the abstract space of states of a quantum mechanical system, and and are any two complex numbers, then is a valid state as well. (Whether the null vector counts as a valid state ("no system present") is a matter of definition. The null vector does ''not'' at any rate describe the vacuum state in quantum field theory.) The set of allowable states is a vector space. This similarity is of course not accidental. There are also a distinctions between the spaces to keep in mind.


Representations

Basic states are characterized by a set of quantum numbers. This is a set of eigenvalues of a maximal set of commuting observables. Physical observables are represented by linear operators, also called observables, on the vectors space. Maximality means that there can be added to the set no further algebraically independent observables that commute with the ones already present. A choice of such a set may be called a choice of representation. * It is a postulate of quantum mechanics that a physically observable quantity of a system, such as position, momentum, or spin, is represented by a linear Hermitian operator on the state space. The possible outcomes of measurement of the quantity are the
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of the operator. At a deeper level, most observables, perhaps all, arise as generators of symmetries.For this statement to make sense, the observables need to be elements of a maximal commuting set. To see this, it is a simple matter to note that, for example, the momentum operator of the i'th particle in a n-particle system is ''not'' a generator of any symmetry in nature. On the other hand, the ''total'' momentum ''is'' a generator of a symmetry in nature; the translational symmetry. * The physical interpretation is that such a set represents what can – in theory – simultaneously be measured with arbitrary precision. The Heisenberg uncertainty relation prohibits simultaneous exact measurements of two non-commuting observables. * The set is non-unique. It may for a one-particle system, for example, be position and spin -projection, , or it may be momentum and spin -projection, . In this case, the operator corresponding to position (a multiplication operator in the position representation) and the operator corresponding to momentum (a
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
in the position representation) do not commute. * Once a representation is chosen, there is still arbitrariness. It remains to choose a coordinate system. This may, for example, correspond to a choice of - and -axis, or a choice of curvilinear coordinates as exemplified by the spherical coordinates used for the Hydrogen atomic wave functions. This final choice also fixes a basis in abstract Hilbert space. The basic states are labeled by the quantum numbers corresponding to the maximal set of commuting observables and an appropriate coordinate system.The resulting basis may or may not technically be a basis in the mathematical sense of Hilbert spaces. For instance, states of definite position and definite momentum are not square integrable. This may be overcome with the use of wave packets or by enclosing the system in a "box". See further remarks below. The abstract states are "abstract" only in that an arbitrary choice necessary for a particular ''explicit'' description of it is not given. This is the same as saying that no choice of maximal set of commuting observables has been given. This is analogous to a vector space without a specified basis. Wave functions corresponding to a state are accordingly not unique. This non-uniqueness reflects the non-uniqueness in the choice of a maximal set of commuting observables. For one spin particle in one dimension, to a particular state there corresponds two wave functions, and , both describing the ''same'' state. * For each choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation that is associated to a function space of wave functions. * Between all these different function spaces and the abstract state space, there are one-to-one correspondences (here disregarding normalization and unobservable phase factors), the common denominator here being a particular abstract state. The relationship between the momentum and position space wave functions, for instance, describing the same state is the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
. Each choice of representation should be thought of as specifying a unique function space in which wave functions corresponding to that choice of representation lives. This distinction is best kept, even if one could argue that two such function spaces are mathematically equal, e.g. being the set of square integrable functions. One can then think of the function spaces as two distinct copies of that set.


Inner product

There is an additional algebraic structure on the vector spaces of wave functions and the abstract state space. * Physically, different wave functions are interpreted to overlap to some degree. A system in a state that does ''not'' overlap with a state cannot be found to be in the state upon measurement. But if overlap to ''some'' degree, there is a chance that measurement of a system described by will be found in states . Also selection rules are observed apply. These are usually formulated in the preservation of some quantum numbers. This means that certain processes allowable from some perspectives (e.g. energy and momentum conservation) do not occur because the initial and final ''total'' wave functions do not overlap. * Mathematically, it turns out that solutions to the Schrödinger equation for particular potentials are orthogonal in some manner, this is usually described by an integral \int\Psi_m^*\Psi_n w\, dV = \delta_, where are (sets of) indices (quantum numbers) labeling different solutions, the strictly positive function is called a weight function, and is the Kronecker delta. The integration is taken over all of the relevant space. This motivates the introduction of an inner product on the vector space of abstract quantum states, compatible with the mathematical observations above when passing to a representation. It is denoted , or in the
Bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
. It yields a complex number. With the inner product, the function space is an inner product space. The explicit appearance of the inner product (usually an integral or a sum of integrals) depends on the choice of representation, but the complex number does not. Much of the physical interpretation of quantum mechanics stems from the Born rule. It states that the probability of finding upon measurement the state given the system is in the state is p = , (\Phi, \Psi), ^2, where and are assumed normalized. Consider a scattering experiment. In quantum field theory, if describes a state in the "distant future" (an "out state") after interactions between scattering particles have ceased, and an "in state" in the "distant past", then the quantities , with and varying over a complete set of in states and out states respectively, is called the S-matrix or scattering matrix. Knowledge of it is, effectively, having ''solved'' the theory at hand, at least as far as predictions go. Measurable quantities such as
decay rate Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
s and scattering cross sections are calculable from the S-matrix.


Hilbert space

The above observations encapsulate the essence of the function spaces of which wave functions are elements. However, the description is not yet complete. There is a further technical requirement on the function space, that of completeness, that allows one to take limits of sequences in the function space, and be ensured that, if the limit exists, it is an element of the function space. A complete inner product space is called a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The property of completeness is crucial in advanced treatments and applications of quantum mechanics. For instance, the existence of projection operators or orthogonal projections relies on the completeness of the space. These projection operators, in turn, are essential for the statement and proof of many useful theorems, e.g. the spectral theorem. It is not very important in introductory quantum mechanics, and technical details and links may be found in footnotes like the one that follows.In technical terms, this is formulated the following way. The inner product yields a norm. This norm, in turn, induces a metric. If this metric is complete, then the aforementioned limits will be in the function space. The inner product space is then called complete. A complete inner product space is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The abstract state space is always taken as a Hilbert space. The matching requirement for the function spaces is a natural one. The Hilbert space property of the abstract state space was originally extracted from the observation that the function spaces forming normalizable solutions to the Schrödinger equation are Hilbert spaces.
The space is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace of . A subspace of a Hilbert space is a Hilbert space if it is closed. In summary, the set of all possible normalizable wave functions for a system with a particular choice of basis, together with the null vector, constitute a Hilbert space. Not all functions of interest are elements of some Hilbert space, say . The most glaring example is the set of functions . These are plane wave solutions of the Schrödinger equation for a free particle, but are not normalizable, hence not in . But they are nonetheless fundamental for the description. One can, using them, express functions that ''are'' normalizable using wave packets. They are, in a sense, a basis (but not a Hilbert space basis, nor a Hamel basis) in which wave functions of interest can be expressed. There is also the artifact "normalization to a delta function" that is frequently employed for notational convenience, see further down. The delta functions themselves aren't square integrable either. The above description of the function space containing the wave functions is mostly mathematically motivated. The function spaces are, due to completeness, very ''large'' in a certain sense. Not all functions are realistic descriptions of any physical system. For instance, in the function space one can find the function that takes on the value for all rational numbers and for the irrationals in the interval . This ''is'' square integrable,As is explained in a later footnote, the integral must be taken to be the Lebesgue integral, the Riemann integral is not sufficient. but can hardly represent a physical state.


Common Hilbert spaces

While the space of solutions as a whole is a Hilbert space there are many other Hilbert spaces that commonly occur as ingredients. * Square integrable complex valued functions on the interval . The set is a Hilbert space basis, i.e. a maximal orthonormal set. * The
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
takes functions in the above space to elements of , the space of ''square summable'' functions . The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces.. This means that inner products, hence norms, are preserved and that the mapping is a bounded, hence continuous, linear bijection. The property of completeness is preserved as well. Thus this is the right concept of isomorphism in the category of Hilbert spaces. Its basis is with . * The most basic example of spanning polynomials is in the space of square integrable functions on the interval for which the Legendre polynomials is a Hilbert space basis (complete orthonormal set). * The square integrable functions on the unit sphere is a Hilbert space. The basis functions in this case are the spherical harmonics. The Legendre polynomials are ingredients in the spherical harmonics. Most problems with rotational symmetry will have "the same" (known) solution with respect to that symmetry, so the original problem is reduced to a problem of lower dimensionality. * The associated Laguerre polynomials appear in the hydrogenic wave function problem after factoring out the spherical harmonics. These span the Hilbert space of square integrable functions on the semi-infinite interval . More generally, one may consider a unified treatment of all second order polynomial solutions to the Sturm–Liouville equations in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well as
Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
, Jacobi polynomials and
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
. All of these actually appear in physical problems, the latter ones in the
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
, and what is otherwise a bewildering maze of properties of special functions becomes an organized body of facts. For this, see . There occurs also finite-dimensional Hilbert spaces. The space is a Hilbert space of dimension . The inner product is the standard inner product on these spaces. In it, the "spin part" of a single particle wave function resides. * In the non-relativistic description of an electron one has and the total wave function is a solution of the Pauli equation. * In the corresponding relativistic treatment, and the wave function solves the Dirac equation. With more particles, the situations is more complicated. One has to employ tensor products and use representation theory of the symmetry groups involved (the
rotation group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
and the Lorentz group respectively) to extract from the tensor product the spaces in which the (total) spin wave functions reside. (Further problems arise in the relativistic case unless the particles are free. See the
Bethe–Salpeter equation The Bethe–Salpeter equation (named after Hans Bethe and Edwin Salpeter) describes the bound states of a two-body (particles) quantum field theoretical system in a relativistically covariant formalism. The equation was actually first publishe ...
.) Corresponding remarks apply to the concept of isospin, for which the symmetry group is SU(2). The models of the nuclear forces of the sixties (still useful today, see nuclear force) used the symmetry group SU(3). In this case, as well, the part of the wave functions corresponding to the inner symmetries reside in some or subspaces of tensor products of such spaces. * In quantum field theory the underlying Hilbert space is Fock space. It is built from free single-particle states, i.e. wave functions when a representation is chosen, and can accommodate any finite, not necessarily constant in time, number of particles. The interesting (or rather the ''tractable'') dynamics lies not in the wave functions but in the
field operator In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible. Historically, this was not quite ...
s that are operators acting on Fock space. Thus the Heisenberg picture is the most common choice (constant states, time varying operators). Due to the infinite-dimensional nature of the system, the appropriate mathematical tools are objects of study in functional analysis.


Simplified description

Not all introductory textbooks take the long route and introduce the full Hilbert space machinery, but the focus is on the non-relativistic Schrödinger equation in position representation for certain standard potentials. The following constraints on the wave function are sometimes explicitly formulated for the calculations and physical interpretation to make sense: * The wave function must be square integrable. This is motivated by the Copenhagen interpretation of the wave function as a probability amplitude. * It must be everywhere continuous and everywhere
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. This is motivated by the appearance of the Schrödinger equation for most physically reasonable potentials. It is possible to relax these conditions somewhat for special purposes.One such relaxation is that the wave function must belong to the Sobolev space ''W''1,2. It means that it is differentiable in the sense of distributions, and its gradient is square-integrable. This relaxation is necessary for potentials that are not functions but are distributions, such as the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
.
If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude. This does not alter the structure of the Hilbert space that these particular wave functions inhabit, but the subspace of the square-integrable functions , which is a Hilbert space, satisfying the second requirement ''is not closed'' in , hence not a Hilbert space in itself.It is easy to visualize a sequence of functions meeting the requirement that converges to a ''discontinuous'' function. For this, modify an example given in Inner product space#Some examples. This element though ''is'' an element of . The functions that does not meet the requirements are still needed for both technical and practical reasons.For instance, in perturbation theory one may construct a sequence of functions approximating the true wave function. This sequence will be guaranteed to converge in a larger space, but without the assumption of a full-fledged Hilbert space, it will not be guaranteed that the convergence is to a function in the relevant space and hence solving the original problem.Some functions not being square-integrable, like the plane-wave free particle solutions are necessary for the description as outlined in a previous note and also further below.


More on wave functions and abstract state space

As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in general
infinite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
Hilbert space. Due to the multiple possible choices of representation basis, these Hilbert spaces are not unique. One therefore talks about an abstract Hilbert space, state space, where the choice of representation and basis is left undetermined. Specifically, each state is represented as an abstract vector in state space. A quantum state in any representation is generally expressed as a vector , \Psi\rangle = \sum_\int d^m\!\boldsymbol\,\, \Psi(\boldsymbol,\boldsymbol,t)\, , \boldsymbol,\boldsymbol\rangle where * the basis vectors of the chosen representation * a " differential volume element" in the continuous degrees of freedom * a component of the vector , called the wave function of the system * dimensionless discrete quantum numbers * continuous variables (not necessarily dimensionless) These quantum numbers index the components of the state vector. More, all are in an -dimensional set where each is the set of allowed values for ; all are in an -dimensional "volume" where and each is the set of allowed values for , a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the real numbers . For generality and are not necessarily equal. Example: The probability density of finding the system at time tat state is \rho_ (t)= , \Psi(\boldsymbol,\boldsymbol,t), ^2 The probability of finding system with in some or all possible discrete-variable configurations, , and in some or all possible continuous-variable configurations, , is the sum and integral over the density,Here: \sum_ \equiv \sum_ \equiv \sum_\sum_\cdots\sum_ is a multiple sum. P(t)=\sum_\int_C \rho_ (t) \,\, d^m\!\boldsymbol Since the sum of all probabilities must be 1, the normalization condition 1=\sum_\int_ \rho_ (t) \, d^m\!\boldsymbol must hold at all times during the evolution of the system. The normalization condition requires to be dimensionless, by dimensional analysis must have the same units as .


Ontology

Whether the wave function really exists, and what it represents, are major questions in the
interpretation of quantum mechanics An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extrao ...
. Many famous physicists of a previous generation puzzled over this problem, such as Schrödinger,
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
and
Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...
. Some advocate formulations or variants of the Copenhagen interpretation (e.g. Bohr, Wigner and von Neumann) while others, such as Wheeler or
Jaynes Jaynes is a surname, and may refer to * Cindy Jaynes (born 1959), American rear-admiral * Dwight Jaynes, American sports journalist * Edwin Thompson Jaynes (1922–1998), American physicist and theorist of probability * Jeremy Jaynes (born 1974), A ...
, take the more classical approach and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, including Schrödinger, Bohm and Everett and others, argued that the wave function must have an objective, physical existence. Einstein thought that a complete description of physical reality should refer directly to physical space and time, as distinct from the wave function, which refers to an abstract mathematical space.


See also


Remarks


Citations


General sources

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Further reading

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External links










Quantum Mechanics and Quantum Computation at BerkeleyX

Einstein, ''The quantum theory of radiation''
{{DEFAULTSORT:Wave Function Quantum states Waves