Editing Wave function (section)

=== Position-space wave functions ===
The state of such a particle is completely described by its wave function, <math display="block">\Psi(x,t)\,,</math> where {{mvar|x}} is position and {{mvar|t}} is time. This is a [[complex-valued function]] of two real variables {{mvar|x}} and {{mvar|t}}.

For one spinless particle in one dimension, if the wave function is interpreted as a [[probability amplitude]]; the square [[absolute value|modulus]] of the wave function, the positive real number
<math display="block"> \left|\Psi(x, t)\right|^2 = \Psi^*(x, t)\Psi(x, t) = \rho(x), </math>
is interpreted as the [[probability density function|probability density]] for a measurement of the particle's position at a given time {{math|''t''}}. The asterisk indicates the [[complex conjugate]]. If the particle's position is [[measurement in quantum mechanics|measured]], its location cannot be determined from the wave function, but is described by a [[probability distribution]].

====Normalization condition====
The probability that its position {{math|''x''}} will be in the interval {{math|''a'' ≤ ''x'' ≤ ''b''}} is the integral of the density over this interval:
<math display="block">P_{a\le x\le b} (t) = \int_a^b \,|\Psi(x,t)|^2 dx </math>
where {{mvar|t}} is the time at which the particle was measured. This leads to the '''normalization condition''':
<math display="block">\int_{-\infty}^\infty \, |\Psi(x,t)|^2dx  = 1\,,</math>
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. Technically, wave functions form a [[Mathematical formulation of quantum mechanics#Description of the state of a system|ray]] in a [[projective Hilbert space]] rather than an ordinary vector space.

====Quantum states as vectors====
{{See also|Mathematical formulation of quantum mechanics|Bra–ket notation|Position operator}}

At a particular instant of time, all values of the wave function {{math|Ψ(''x'', ''t'')}} are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In [[Bra–ket notation]], this vector is written
<math display="block">|\Psi(t)\rangle = \int\Psi(x,t) |x\rangle  dx </math>
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 (linear algebra)|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]] can be used to manipulate wave functions.
* The idea that [[quantum state]]s are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and [[quantum field theory]], 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 {{mvar|x}} coordinate is a continuous index. The {{math|{{ket|''x''}}}} are called ''improper vectors'' which, unlike ''proper vectors'' that are normalizable to unity, can only be normalized to a Dirac delta function.<ref group="nb">As, technically, they are not in the Hilbert space. See [[Self-adjoint operator#Spectral theorem|Spectral theorem]] for more details.</ref><ref name=":0" group="nb" />{{sfn|Shankar|1994|p=117}}
<math display="block">\langle x' | x \rangle = \delta(x' - x) </math>
thus
<math display="block">\langle x' |\Psi\rangle = \int \Psi(x) \langle x'|x\rangle dx= \Psi(x') </math>
and
<math display="block">|\Psi\rangle = \int  |x\rangle \langle x |\Psi\rangle dx= \left( \int |x\rangle \langle x |dx\right) |\Psi\rangle </math>
which illuminates the [[identity operator]]
<math display="block">I = \int |x\rangle \langle x | dx\,. </math>which is analogous to completeness relation of orthonormal basis in N-dimensional Hilbert space.

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).