Noun
 mathematics physics A mathematical function that describes the propagation of the quantum mechanical wave associated with a particle (or system of particles); related to the probability of finding the particle in a particular region of space
Related terms
Translations
 Dutch: Golffunctie
 German: Wellenfunktion
 Japanese: 波動関数
 Russian: Волновая функция
A wave function or wavefunction is a mathematical
tool used in quantum
mechanics to describe any physical system. It is a function
from a space
that maps the possible states of the system into the complex
numbers. The laws of quantum mechanics (i.e. the Schrödinger
equation) describe how the wave function evolves over time. The
values of the wave function are probability
amplitudes — complex numbers — the squares of the absolute
values of which, give the probability
distribution that the system will be in any of the possible
states.
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.
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:
 a complex vector with finitely many components
 \vec \psi = \begin c_1 \\ \vdots \\ c_n \end,
 a complex vector with infinitely many components
 \vec \psi = \begin c_1 \\ \vdots \\ c_n \\ \vdots \end,
where each component may be complex function of
one or more real
variables (a "continuously indexed" complex vector)
 \psi(x_1, \, \ldots \, x_n).
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: 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:
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
\psi(x)\, defined over the real line. The
positive function \psi^2\, 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
 \mathbf_ = \int_^ \psi(x)^2\, dx .
This leads to the normalization
condition
 \int_^ \psi(x)^2\, dx = 1 \quad .
since the probability of a measurement of the
particle's position yielding a value in the range (\infty, \infty)
is unity.
One particle in three spatial dimensions
The
three dimensional case is analogous to the one dimensional
case; the wave function is a complex function \psi(x, y, z)\,
defined over three dimensional space, and its complex square is
interpreted as a three dimensional probability density
function:
 \mathbf_R = \int_R \psi(x, y, z)^2 \, dV
The normalization condition is likewise
 \int_R \psi(x, y, z)^2\, dV = 1
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, \psi(x_1, y_1, z_1, x_2, y_2,
z_2) \ , and \psi^2\, 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
 \mathbf_ = \int_R \int_S \psi^2 \, dV_2 \, dV_1
where dV_1 = dx_1 dy_1 dz_1, and similarly for
dV_2.
The normalization condition is then:
 \int \int \psi(x, y, z)^2 \, dV_2 \, dV_1 = 1
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 singleparticle subsystem. In other
words, the particles in the system can be entangled.
One particle in one dimensional momentum space
The wave function for a one dimensional particle
in momentum space is a complex function \psi(p)\, defined over the
real
line. The quantity \psi^2\, is interpreted as a probability
density function in momentum
space:
 \mathbf_ = \int_^ \psi(p)^2\, dp
As in the position space case, this leads to the
normalization condition:
 \int_^ \psi(p)^2\, dp = 1 .
Spin 1/2
The wave function for a spin½ particle
(ignoring its spatial degrees of freedom) is a column vector
 \vec \psi = \begin c_1 \\ c_2 \end.
The meaning of the vector's components depends on
the basis, but typically c_1 and c_2 are respectively the
coefficients of spin up and spin down in the z direction. In
Dirac
notation this is:
  \psi \rangle = c_1  \uparrow_z \rangle + c_2  \downarrow_z \rangle
The values c_1^2 \, and c_2^2 \, are then
respectively interpreted as the probability of obtaining spin up or
spin down in the z direction when a measurement of the particle's
spin is performed. This leads to the normalization condition
 c_1^2 + c_2^2 = 1\,.
Interpretation
A wave function describes the state of a physical
system,  \psi \rangle\,, by expanding it in terms of other
possible states of the same system,  \phi_i \rangle. Collectively
the latter are referred to as a basis or representation. In what
follows, all wave functions are assumed to be normalized.
Finite dimensional basis vectors
A wave function which is a vector \vec \psi with
n components describes how to express the state of the physical
system  \psi \rangle as the linear combination of finitely many
basis elements  \phi_i \rangle, where i runs from 1 to n. In
particular the equation
 \vec \psi = \begin c_1 \\ \vdots \\ c_n \end,
which is a relation between column vectors, is
equivalent to
 \psi \rangle = \sum_^n c_i  \phi_i \rangle,
which is a relation between the states of a
physical system. Note that to pass between these expressions one
must know the basis in use, and hence, two column vectors with the
same components can represent two different states of a system if
their associated basis states are different. An example of a wave
function which is a finite vector is furnished by the spin state of
a spin1/2 particle, as described above.
The physical meaning of the components of \vec
\psi is given by the wave function collapse postulate:
 If the states  \phi_i \rangle have distinct, definite values, \lambda_i, of some dynamical variable (e.g. momentum, position, etc) and a measurement of that variable is performed on a system in the state

 \psi \rangle = \sum_i c_i  \phi_i \rangle
 then the probability of measuring \lambda_i is c_i^2, and if the measurement yields \lambda_i, the system is left in the state  \phi_i \rangle.
Infinite dimensional basis vectors
The case of an infinite vector with a discrete
index is treated in the same manner a finite vector, except the sum
is extended over all the basis elements. Hence
 \vec \psi = \begin c_1 \\ \vdots \\ c_n \\ \vdots \end
is equivalent to
 \psi \rangle = \sum_ c_i  \psi_i \rangle,
where it is understood that the above sum
includes all the components of \vec \psi. The interpretation of the
components is the same as the finite case (apply the collapse
postulate).
Continuously indexed vectors (functions)
In the case of a continuous index, the sum is
replaced by an integral; an example of this is the spatial wave
function of a particle in one dimension, which expands the physical
state of the particle,  \psi \rangle, in terms of states with
definite position,  x \rangle. Thus
  \psi \rangle = \int_^ \psi(x)  x \rangle\,dx.
Note that  \psi \rangle is not the same as
\psi(x)\,. The former is the actual state of the particle, whereas
the latter is simply a wave function describing how to express the
former as a superposition of states with definite position. In this
case the base states themselves can be expressed as
  x_0 \rangle = \int_^ \delta(x  x_0)  x \rangle\,dx
and hence the spatial wave function associated
with  x_0 \rangle is \delta(x  x_0)\, (where \delta(x  x_0)\, is
the Dirac
delta function).
Formalism
Given an isolated physical system, the allowed
states of this system (i.e. the states the system could occupy
without violating the laws of physics) are part of a Hilbert
space H. Some properties of such a space are
 1. If  \psi \rangle and  \phi \rangle are two allowed states, then


 a  \psi \rangle + b  \phi \rangle

 is also an allowed state, provided a^2+b^2=1. (This condition is due to normalisation.)
 2. There is always an orthonormal basis of allowed states of the vector space H.
The wave function associated with a particular
state may be seen as an expansion of the state in a basis of H. For
example,
 \
is a basis for the space associated with the spin
of a spin1/2 particle and consequently the spin state of any such
particle can be written uniquely as
 a\uparrow_z \rangle + b\downarrow_z \rangle.
Sometimes it is useful to expand the state of a
physical system in terms of states which are not allowed, and
hence, not in H. An example of this is the spacial wave function
associated with a particle in one dimension which expands the state
of the particle in terms of states with definite position.
Every Hilbert space H is equipped with an
inner
product. Physically, the nature of the inner product is
contingent upon the kind of basis in use. When the basis is a
countable set \\,, and orthonormal, i.e.
 \langle \phi_i  \phi_j \rangle = \delta_.
Then an arbitrary vector  \psi \rangle can be
expressed as
  \psi \rangle = \sum_i c_i  \phi_i \rangle
where c_i = \langle \phi_i  \psi \rangle.
If one chooses a "continuous" basis as, for
example, the position or coordinate basis consisting of all states
of definite position \, the orthonormality condition holds
similarly:
 \langle x  x' \rangle = \delta(x  x').
We have the analogous identity
 \langle x  \int \psi(x')  x' \rangle \,dx' = \int \psi(x') \delta(x  x')\,dx' = \psi(x).
Ontology
Whether the wave function is real, and what it
represents, are major questions in the
interpretation of quantum mechanics. Many famous physicists
have puzzled over this problem, such as Erwin
Schrödinger, Albert
Einstein and Niels Bohr.
Some approaches regard it as merely representing information in the
mind of the observer. Others argue that it must be objective.
Notes
See also
References
 Introduction to Quantum Mechanics (2nd ed.)
wavefunction in Bosnian: Talasna funkcija
wavefunction in Czech: Vlnová funkce
wavefunction in German: Wellenfunktion
wavefunction in Modern Greek (1453):
Κυματοσυνάρτηση
wavefunction in Spanish: Función de ondas
wavefunction in Persian: تابع موج
wavefunction in Finnish: Aaltofunktio
wavefunction in French: Fonction d'onde
wavefunction in Galician: Función de onda
wavefunction in Hebrew: פונקציית גל
wavefunction in Hungarian: Hullámfüggvény
wavefunction in Italian: Funzione d'onda
wavefunction in Japanese: 波動関数
wavefunction in Korean: 파동함수
wavefunction in Lithuanian: Banginė
funkcija
wavefunction in Dutch: Golffunctie
wavefunction in Norwegian: Bølgefunksjon
wavefunction in Polish: Funkcja falowa
wavefunction in Portuguese: Função de onda
wavefunction in Romanian: Funcţie de undă
wavefunction in Russian: Волновая функция
wavefunction in Slovak: Vlnová funkcia
wavefunction in Slovenian: Valovna
funkcija
wavefunction in Swedish: Vågfunktion
wavefunction in Turkish: Dalga fonksiyonu
wavefunction in Ukrainian: Хвильова
функція
wavefunction in Vietnamese: Hàm sóng
wavefunction in Chinese: 波函数