This article discusses the concept of a wavefunction as it relates to quantum mechanics. The term has a significantly different meaning when used in the context of classical mechanics or classical electromagnetism. A simple introduction to this subject is provided in Basics of quantum mechanics. ...
Definition
The modern usage of the term wavefunction refers to any vector or function which describes the state of a physical system by expanding it in terms of other states of the same system. Typically, a wavefunction is either: A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
Partial plot of a function f. ...
A physical system is a system that is comprised of matter and energy. ...
 a complex vector with finitely many components
 ,
 a complex vector with infinitely many components
 ,
 or a complex function of one or more real variables (a "continuously indexed" complex vector)
 .
In all cases, the wavefunction provides a complete description of the associated physical system. However, it is important to note that the wavefunction associated with a system is not uniquely determined by that system, as many different wavefunctions may describe the same physical scenario. The complex numbers are an extension of the real numbers, in which all nonconstant polynomials have roots. ...
In mathematics, the real numbers are intuitively defined as numbers that are in onetoone correspondence with the points on an infinite lineâ€”the number line. ...
Interpretation The physical interpretation of the wavefunction 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 wavefunction associated with a particle in one dimension is a complex function defined over the real line. The complex square of the wavefunction, , is interpreted as the probability density associated with the particle's position, and hence the probability that a measurement of the particle's position yields a value in the interval [a,b] is The complex numbers are an extension of the real numbers, in which all nonconstant polynomials have roots. ...
Partial plot of a function f. ...
In quantum mechanics, a probability amplitude is a complex numbervalued function which describes an uncertain or unknown quantity. ...
 .
This leads to the normalization condition In quantum mechanics, wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must equal 1. ...
 .
since a measurement of the particle's position must produce a real number.
One particle in three spatial dimensions The three dimensional case is analogous to the one dimensional case; the wavefunction is a complex function defined over three dimensional space, and its complex square is interpreted as a three dimensional probability density function. The probability that a measurement of the particle's position results in a value which is in the volume R is thus Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
 .
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 wavefunction is a complex function of six spatial variables,  ,
and is a joint probability density function associated with the positions of both particles. 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 then where dV_{1} = dx_{1}dy_{1}dz_{1} and similarly for dV_{2}. The normalization condition is thus where the preceding integral is taken over the full range of all six variables. It is of crucial importance to realize that, in the case of two particle systems, only the system consisting of both particles need have a well defined wavefunction. That is, it may be impossible to write down a probability density function for the position of particle one which does not depend explicitly on the position of particle two. This gives rise to the phenomenon of quantum entanglement. Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ...
One particle in one dimensional momentum space The wavefunction for a one dimensional particle in momentum space is a complex function defined over the real line. The quantity is interpreted as a probability density function in momentum space, and hence the probability that a measurement of the particle's momentum yields a value in the interval [a,b] is  .
This leads to the normalization condition since a measurement of the particle's momentum always results in a real number.
Spin 1/2 The wavefunction for a spin 1/2 particle (ignoring its spacial degrees of freedom) is a column vector  .
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: Braket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...
The values and 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  .
Interpretation A wavefunction describes the state of a physical system by expanding it in terms of other states of the same system. We shall denote the state of the system under consideration as and the states into which it is being expanded as . Collectively the latter are referred to as a basis or representation. In what follows, all wavefunctions are assumed to be normalized.
Finite vectors A wavefunction which is a vector with n components describes how to express the state of the physical system as the linear combination of finitely many basis elements , where i runs from 1 to n. In particular the equation  ,
which is a relation between column vectors, is equivalent to  ,
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 wavefunction 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 is given by the wavefunction collapse postulate:  If the states have distinct, definite values, λ_{i}, of some dynamical variable (e.g. momentum, position, etc) and a measurement of that variable is performed on a system in the state
 then the probability of measuring λ_{i} is  c_{i}  ^{2}, and if the measurement yields λ_{i}, the system is left in the state .
Infinite 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 is equivalent to  ,
where it is understood that the above sum includes all the components of . 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 wavefunction of a particle in one dimension, which expands the physical state of the particle, , in terms of states with definite position, . Thus  .
Note that is not the same as . The former is the actual state of the particle, whereas the latter is simply a wavefunction 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 and hence the spatial wavefunction associated with is .
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 vector space H, the Hilbert space. That is, A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...
 1. If and are two allowed states, then


 is also an allowed state, provided a^{2} + b^{2} = 1. (This condition is due to normalisation.)
and,  2. Due to normalisation, there is always an orthonormal basis of allowed states of the vector space H.
In this context the wavefunction associated with a particular state may be seen as an expansion of the state in a basis for the vector space H. For example, In mathematics, an orthonormal basis of an inner product space V(i. ...
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  .
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 wavefunction associated with a particle in one dimension which expands the state of the particle in terms of states with definite position. These states are forbidden, however, since they violate the uncertainty principle. Bases such as these as called improper bases. It is conventional to endow H with an inner product but the nature of the inner product is contingent upon the kind of basis in use. When there are countably many basis elements all of which belong to H, H is equipped with the unique inner product that makes this basis orthornormal, i.e., // Definition Inner Product of two vectors Given twoNby1 column vectors v and u, the inner product is defined as the scalar quantity Î± resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...
When this is done, the inner product of with the expansion of an arbitrary vector is  .
If the basis elements constitute a continuum, as, for example, the position or coordinate basis consisting of all states of definite position , it is conventional to choose the Dirac normalization so that the analogous identity  .
holds.
See also The wave packet is one of the most widely misunderstood and misused concepts in physics. ...
In physics, bosons, named after Satyendra Nath Bose, are particles with integer spin. ...
Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...
In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...
In particle physics, fermions, (named after Enrico Fermi), are particles with semiinteger spin. ...
In set theory, the adjective antisymmetric usually refers to an antisymmetric relation. ...
A simple introduction to this subject is provided in Basics of quantum mechanics. ...
In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the timedependence of quantum mechanical systems. ...
In quantum mechanics, wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must equal 1. ...
References  Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.), Prentice Hall. ISBN 013805326X.
