Editing Wave function (section)

== Definition (one spinless particle in one dimension) ==
{{Unreferenced section|date=May 2021}}
{{multiple image
 | align     = right
 | direction = vertical
 | width     = 402
 | footer    = The [[Real and imaginary parts|real parts]] of position wave function {{math|Ψ(''x'')}} and momentum wave function {{math|Φ(''p'')}}, and corresponding probability densities {{math|{{!}}Ψ(''x''){{!}}<sup>2</sup>}} and {{math|{{!}}Φ(''p''){{!}}<sup>2</sup>}}, for one spin-0 particle in one {{mvar|x}} or {{mvar|p}} dimension. The colour opacity of the particles corresponds to the probability density (''not'' the wave function) of finding the particle at position {{mvar|x}} or momentum {{math|''p''}}.
 | image1    = Quantum mechanics standing wavefunctions.svg
 | caption1  = [[Standing wave]]s for a [[particle in a box]], examples of [[stationary state]]s.
 | image2    = Quantum mechanics travelling wavefunctions.svg
 | caption2  = Travelling waves of a free particle.
}}

For now, consider the simple case of a non-relativistic single particle, without [[Spin (physics)|spin]], in one spatial dimension. More general cases are discussed below.

According to the [[postulates of quantum mechanics]], the [[Quantum state|state]] of a physical system, at fixed time <math>t</math>, is given by the wave function belonging to a [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]].{{sfn |Applications of Quantum Mechanics}}{{sfn | Griffiths | 2004 | p=94}} As such, the [[inner product]] of two wave functions {{math|Ψ<sub>1</sub>}} and {{math|Ψ<sub>2</sub>}} can be defined as the complex number (at time {{mvar|t}})<ref group="nb">The functions are here assumed to be elements of {{math|[[Lp-space|''L''<sup>2</sup>]]}}, 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]] {{math|0}}. This is necessary to obtain an inner product (that is, {{math|1=(Ψ, Ψ) = 0 ⇒ Ψ ≡ 0}}) 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.</ref>
:<math>( \Psi_1 , \Psi_2 ) = \int_{-\infty}^\infty \, \Psi_1^*(x, t)\Psi_2(x, t)\,dx < \infty</math>.

More details are given [[Wave function#Wave functions and function spaces|below]]. However,  the inner product of a wave function {{math|Ψ}} with itself,
:<math>(\Psi,\Psi) = \|\Psi\|^2</math>,
is ''always'' a positive real number. The number {{math|{{norm|Ψ}}}} (not {{math|{{norm|Ψ}}<sup>2</sup>}}) is called the '''[[norm (mathematics)|norm]]''' of the wave function {{math|Ψ}}.
The [[Hilbert space#Separable spaces|separable Hilbert space]] being considered is infinite-[[Dimension (vector space)|dimensional]],<ref group="nb">In quantum mechanics, only [[Hilbert space#Separable spaces|separable Hilbert spaces]] are considered, which using [[Zorn's lemma|Zorn's Lemma]], implies it admits a countably infinite [[Schauder basis]] rather than an orthonormal basis in the sense of linear algebra ([[Basis (linear algebra)#Hamel basis|Hamel basis]]).</ref> which means there is no finite set of [[Square-integrable function|square integrable functions]] which can be added together in various combinations to create every possible [[Square-integrable function|square integrable function]].

=== 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).

=== Momentum-space wave functions ===
The particle also has a wave function in [[momentum space]]:
<math display="block">\Phi(p,t)</math>
where {{mvar|p}} is the [[Momentum#Quantum mechanical|momentum]] in one dimension, which can be any value from {{math|−∞}} to {{math|+∞}}, and {{mvar|t}} is time.

Analogous to the position case, the inner product of two wave functions {{math|Φ<sub>1</sub>(''p'', ''t'')}} and {{math|Φ<sub>2</sub>(''p'', ''t'')}} can be defined as:
<math display="block">(\Phi_1 , \Phi_2 ) = \int_{-\infty}^\infty \, \Phi_1^*(p, t)\Phi_2(p, t) dp\,.</math>

One particular solution to the time-independent Schrödinger equation is
<math display="block">\Psi_p(x) = e^{ipx/\hbar},</math>
a [[plane wave]], which can be used in the description of a particle with momentum exactly {{mvar|p}}, since it is an eigenfunction of the [[momentum operator]]. These functions are not normalizable to unity (they are not square-integrable), so they are not really elements of physical Hilbert space. The set
<math display="block">\{\Psi_p(x, t), -\infty \le p \le \infty\}</math>
forms what is called the '''momentum basis'''. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions are not normalizable, they are instead '''normalized to a delta function''',<ref group="nb" name=":0">Also called "Dirac orthonormality", according to {{cite book | last = Griffiths | first = David J. | title = Introduction to Quantum Mechanics | edition = 3rd}}</ref>
<math display="block">(\Psi_{p},\Psi_{p'}) = \delta(p - p').</math>

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 {{math|''x''}} and {{math|''p''}} representations are
<math display="block">\begin{align}
|\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{align}</math>

Now take the projection of the state {{math|Ψ}} onto eigenfunctions of momentum using the last expression in the two equations,
<math display="block">\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).</math>

Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the [[Free_particle#Mathematical_description|free Schrödinger equation]]
<math display="block">\langle x | p \rangle = p(x) = \frac{1}{\sqrt{2\pi\hbar}}e^{\frac{i}{\hbar}px} \Rightarrow \langle p | x \rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-\frac{i}{\hbar}px},</math>
one obtains
<math display="block">\Phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \Psi(x)e^{-\frac{i}{\hbar}px}dx\,.</math>

Likewise, using eigenfunctions of position,
<math display="block">\Psi(x) = \frac{1}{\sqrt{2\pi\hbar}}\int \Phi(p)e^{\frac{i}{\hbar}px}dp\,.</math>

The position-space and momentum-space wave functions are thus found to be [[Fourier transform]]s of each other.{{sfn|Griffiths|2004}} They are two representations of the same state; containing the same information, and either one is sufficient to calculate any property of the particle.

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]], {{mvar|x}} and {{mvar|p}} enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in {{math|''L''<sup>2</sup>}}.<ref group=nb>The Fourier transform viewed as a unitary operator on the space {{math|''L''<sup>2</sup>}} has eigenvalues {{math|±1, ±''i''}}. The eigenvectors are "Hermite functions", i.e. [[Hermite polynomials]] multiplied by a [[Gaussian function]]. See {{harvtxt|Byron|Fuller|1992}} for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.</ref>