Editing Wave function (section)

==Non-relativistic examples==
The following are solutions to the Schrödinger equation for one non-relativistic spinless particle.

===Finite potential barrier===
[[File:Finitepot.png|thumb|Scattering at a finite potential barrier of height {{math|''V''<sub>0</sub>}}. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. {{math|''E'' > ''V''<sub>0</sub>}} for this illustration.]]

One of the most prominent features of wave mechanics is the possibility for a particle to reach a location with a prohibitive (in classical mechanics) [[potential energy|force potential]]. A common model is the "[[potential barrier]]", the one-dimensional case has the potential
<math display="block">V(x)=\begin{cases}V_0 & |x|<a \\ 0 & | x | \geq a\end{cases}</math>
and the steady-state solutions to the wave equation have the form (for some constants {{math|''k'', ''κ''}})
<math display="block">\Psi (x) = \begin{cases}
A_{\mathrm{r}}e^{ikx}+A_{\mathrm{l}}e^{-ikx} & x<-a, \\
B_{\mathrm{r}}e^{\kappa x}+B_{\mathrm{l}}e^{-\kappa x} & |x|\le a, \\
C_{\mathrm{r}}e^{ikx}+C_{\mathrm{l}}e^{-ikx} & x>a.
\end{cases}</math>

Note that these wave functions are not normalized; see [[scattering theory]] 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 {{mvar|x}}): setting {{math|1=''A''<sub>r</sub> = 1}} corresponds to firing particles singly; the terms containing {{math|''A''<sub>r</sub>}} and {{math|''C''<sub>r</sub>}} signify motion to the right, while {{math|''A''<sub>l</sub>}} and {{math|''C''<sub>l</sub>}} – to the left. Under this beam interpretation, put {{math|1=''C''<sub>l</sub> = 0}} 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.

[[File:Quantum dot.png|thumb|upright=1.3|3D confined electron wave functions in a quantum dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ''s-type'' and ''p-type''. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation)|link=File:QuantumDot_wf.gif]]

In a semiconductor [[crystallite]] whose radius is smaller than the size of its [[exciton]] [[Bohr radius]], the excitons are squeezed, leading to [[Potential well#Quantum confinement|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]] can be expressed in terms of [[Hermite polynomial]]s {{math|''H<sub>n</sub>''}}, they are
<math display="block">  \Psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{
- \frac{m\omega x^2}{2 \hbar}} \cdot H_n{\left(\sqrt{\frac{m\omega}{\hbar}} x \right)} </math>
where {{math|1=''n'' = 0, 1, 2, ...}}.

[[File:Hydrogen Density Plots.png|thumb|upright=1.5|The electron probability density for the first few [[hydrogen atom]] electron [[atomic orbital|orbitals]] shown as cross-sections. These orbitals form an [[orthonormal basis]] for the wave function of the electron. Different orbitals are depicted with different scale.]]

===Hydrogen atom===
The wave functions of an electron in a [[Hydrogen atom#Mathematical summary of eigenstates of hydrogen atom|Hydrogen atom]] are expressed in terms of [[spherical harmonics]] and [[Laguerre polynomial#Generalized Laguerre polynomials|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,<ref>Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, {{ISBN|0-7167-8964-7}}</ref>
<math display="block"> \Psi_{n\ell m}(r,\theta,\phi) = R(r)\,\,Y_\ell^m\!(\theta, \phi)</math>
where {{math|''R''}} are radial functions and {{math|''Y''{{su|p=''m''|b=''ℓ''}}(''θ'', ''φ'')}} are [[spherical harmonic]]s of degree {{math|''ℓ''}} and order {{math|''m''}}. 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:{{sfn|Griffiths|2008|pp=162ff}}
<math display="block"> \Psi_{n\ell m}(r,\theta,\phi) = \sqrt {{\left (  \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^{\ell} L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^{m}(\theta, \phi ) </math>
where {{math|1=''a''<sub>0</sub> = 4''πε''<sub>0</sub>''ħ''<sup>2</sup>/''m<sub>e</sub>e''<sup>2</sup>}} is the [[Bohr radius]],
{{math|''L''{{su|b=''n'' − ''ℓ'' − 1|p=2''ℓ'' + 1}}}} are the [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] of degree {{math|''n'' − ''ℓ'' − 1}}, {{math|1=''n'' = 1, 2, ...}} is the [[principal quantum number]], {{math|1=''ℓ'' = 0, 1, ..., ''n'' − 1}} the [[azimuthal quantum number]], {{math|1=''m'' = −''ℓ'', −''ℓ'' + 1, ...,  ''ℓ'' − 1, ''ℓ''}} the [[magnetic quantum number]]. [[Hydrogen-like atom]]s 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 {{math|(''n'', ''ℓ'', ''m'')}}, 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 {{math|[[Lp space|''L''<sup>2</sup>]]}}.
* The displayed functions are solutions to the Schrödinger equation. Obviously, not every function in {{math|''L''<sup>2</sup>}} satisfies the Schrödinger equation for the hydrogen atom. The function space is thus a subspace of {{math|''L''<sup>2</sup>}}.
* The displayed functions form part of a basis for the function space. To each triple {{math|(''n'', ''ℓ'', ''m'')}}, 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]].
* The basis functions are mutually [[orthonormal]].

==Wave functions and function spaces==
The concept of [[function space]]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]] on the set (in the present case a [[vector space]] structure with an [[inner product]]), together with a [[Topological space|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 set|closed]]. It will be concluded below that the function space of wave functions is a [[Hilbert space]]. 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 {{math|Ψ}} and {{math|Φ}} are two states in the abstract space of '''states''' of a quantum mechanical system, and {{math|''a''}} and {{math|''b''}} are any two complex numbers, then {{math|''a''Ψ + ''b''Φ}} 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 [[Canonical commutation relation|commuting]] [[observable]]s. 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 and eigenvectors|eigenvalues]] of the operator.{{sfn|Weinberg|2013}} At a deeper level, most observables, perhaps all, arise as generators of [[Symmetry in quantum mechanics|symmetries]].{{sfn|Weinberg|2013}}{{sfn|Weinberg|2002}}<ref group=nb>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.</ref>
* 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 {{math|''z''}}-projection, {{math|(''x'', ''S''<sub>''z''</sub>)}}, or it may be momentum and spin {{math|''y''}}-projection, {{math|(''p'', ''S''<sub>''y''</sub>)}}. 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 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 {{math|''x'', ''y''}}- and {{math|''z''}}-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.<ref group=nb>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 packet]]s or by enclosing the system in a "box". See further remarks below.</ref>

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, {{math|Ψ(''x'', ''S''<sub>''z''</sub>)}} and {{math|Ψ(''p'', ''S''<sub>''y''</sub>)}}, 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]].

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 {{math|Ψ}} that does ''not'' overlap with a state {{math|Φ}} cannot be found to be in the state {{math|Φ}} upon measurement. But if {{math|Φ<sub>1</sub>, Φ<sub>2</sub>, …}} overlap {{math|Ψ}} to ''some'' degree, there is a chance that measurement of a system described by {{math|Ψ}} will be found in states {{math|Φ<sub>1</sub>, Φ<sub>2</sub>, …}}. Also [[selection rule]]s 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 <math display="block">\int\Psi_m^*\Psi_n w\, dV = \delta_{nm},</math> where {{math|''m'', ''n''}} are (sets of) indices (quantum numbers) labeling different solutions, the strictly positive function {{mvar|w}} is called a weight function, and {{math|''δ''<sub>''mn''</sub>}} 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 {{math|(Ψ, Φ)}}, or in the [[Bra–ket notation]] {{math|{{braket|bra-ket|Ψ|Φ}}}}. 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 {{math|(Ψ, Φ)}} does not. Much of the physical interpretation of quantum mechanics stems from the [[Born rule]]. It states that the probability {{mvar|p}} of finding upon measurement the state {{math|Φ}} given the system is in the state {{math|Ψ}} is
<math display="block">p = |(\Phi, \Psi)|^2,</math>
where {{math|Φ}} and {{math|Ψ}} are assumed normalized. Consider a [[Scattering theory|scattering experiment]]. In quantum field theory, if {{math|Φ<sub>out</sub>}} describes a state in the "distant future" (an "out state") after interactions between scattering particles have ceased, and {{math|Ψ<sub>in</sub>}} an "in state" in the "distant past", then the quantities {{math|(Φ<sub>out</sub>, Ψ<sub>in</sub>)}}, with {{math|Φ<sub>out</sub>}} and {{math|Ψ<sub>in</sub>}} 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]]s and [[scattering cross section]]s are calculable from the S-matrix.{{sfn|Weinberg|2002|loc=Chapter 3}}

=== 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 [[Complete metric space|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]]. The property of completeness is crucial in advanced treatments and applications of quantum mechanics. For instance, the existence of [[projection operator]]s or '''orthogonal projections''' relies on the completeness of the space.{{sfn|Conway|1990}} 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.<ref group=nb>In technical terms, this is formulated the following way. The inner product yields a [[Normed vector space|norm]]. This norm, in turn, induces a [[Metric space|metric]]. If this metric is [[Complete metric|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]]. 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.</ref>
The space {{math|''L''<sup>2</sup>}} is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace of {{math|''L''<sup>2</sup>}}. 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 {{math|''L''<sup>2</sup>}}. The most glaring example is the set of functions {{math|''e''<sup>{{frac|2''πi'''''p''' · '''x'''|h}}</sup>}}. These are plane wave solutions of the Schrödinger equation for a [[free particle]] that are not normalizable, hence not in {{math|''L''<sup>2</sup>}}. But they are nonetheless fundamental for the description. One can, using them, express functions that ''are'' normalizable using [[wave packet]]s. 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 are not 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 {{math|''L''<sup>2</sup>}} one can find the function that takes on the value {{math|0}} for all rational numbers and {{math|-''i''}} for the irrationals in the interval {{math|[0, 1]}}. This ''is'' square integrable,<ref group=nb>As is explained in a later footnote, the integral must be taken to be the [[Lebesgue integral]], the [[Riemann integral]] is not sufficient.</ref>
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 {{closed-closed|0, 2''π''}}. The set {{math|{''e''<sup>''int''</sup>/2''π'', ''n'' ∈ '''Z'''} }} is a Hilbert space basis, i.e. a maximal orthonormal set.
* The [[Fourier transform]] takes functions in the above space to elements of {{math|''l''<sup>2</sup>('''Z''')}}, the space of ''square summable'' functions {{math|'''Z''' → '''C'''}}. The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces.<ref group=nb>{{harvnb|Conway|1990}}. 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 theory|category]] of Hilbert spaces.</ref> Its basis is {{math|{''e''<sub>''i''</sub>, ''i'' ∈ '''Z'''}<nowiki/>}} with {{math|1=''e''<sub>''i''</sub>(''j'') = ''δ''<sub>''ij''</sub>, ''i'', ''j'' ∈ '''Z'''}}.
* The most basic example of spanning polynomials is in the space of square integrable functions on the interval {{closed-closed|–1, 1}} for which the [[Legendre polynomials]] is a Hilbert space basis (complete orthonormal set).
* The square integrable functions on the [[unit sphere]] {{math|''S''<sup>2</sup>}} 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 [[Laguerre polynomials|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 {{closed-open|0, ∞}}.

More generally, one may consider a unified treatment of all second order polynomial solutions to the [[Sturm–Liouville theory|Sturm–Liouville equations]] in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well as [[Chebyshev polynomials]], [[Jacobi polynomials]] and [[Hermite polynomials]]. All of these actually appear in physical problems, the latter ones in the [[Harmonic oscillator (quantum)|harmonic oscillator]], and what is otherwise a bewildering maze of properties of [[special functions]] becomes an organized body of facts. For this, see {{harvtxt|Byron|Fuller|1992|loc=Chapter 5}}.

There occurs also finite-dimensional Hilbert spaces. The space {{math|'''C'''<sup>''n''</sup>}} is a Hilbert space of dimension {{mvar|n}}. 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 {{math|1=''n'' = 2}} and the total wave function is a solution of the [[Pauli equation]].
* In the corresponding relativistic treatment, {{math|1=''n'' = 4}} and the wave function solves the [[Dirac equation]].

With more particles, the situations is more complicated. One has to employ [[tensor product]]s and use representation theory of the symmetry groups involved (the [[rotation group]] 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.{{sfn|Greiner|Reinhardt|2008}} See the [[Bethe–Salpeter equation]].) 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 {{math|'''C'''<sup>''n''</sup>}} 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]]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 ===
[[File:wavefunction continuity space.svg|thumb|Continuity of the wave function and its first spatial derivative (in the ''x'' direction, ''y'' and ''z'' coordinates not shown), at some time ''t''.]]
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:{{sfn|Eisberg|Resnick|1985}}{{sfn|Rae|2008}}

* 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 function|continuous]] and everywhere [[continuously differentiable]]. 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.<ref group=nb>One such relaxation is that the wave function must belong to the [[Sobolev space]] ''W''<sup>1,2</sup>. It means that it is differentiable in the sense of [[Distribution (mathematics)|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]].</ref>
If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude.{{sfn|Atkins|1974|p=258}} Note that exceptions can arise to the continuity of derivatives rule at points of infinite discontinuity of potential field. For example, in [[particle in a box]] where the derivative of wavefunction can be discontinuous at the boundary of the box where the potential is known to have infinite discontinuity.

This does not alter the structure of the Hilbert space that these particular wave functions inhabit, but the subspace of the square-integrable functions {{math|''L''<sup>2</sup>}}, which is a Hilbert space, satisfying the second requirement ''is not closed'' in {{math|''L''<sup>2</sup>}}, hence not a Hilbert space in itself.<ref group=nb>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 {{math|''L''<sup>2</sup>}}.</ref>
The functions that does not meet the requirements are still needed for both technical and practical reasons.<ref group=nb>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.</ref><ref group=nb>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.</ref>

== More on wave functions and abstract state space ==
{{main|Quantum state}}
As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in general [[Dimension (vector space)|infinite-dimensional]] 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.{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|pp=103,215}} A quantum state {{math|{{ket|Ψ}}}} in any representation is generally expressed as a vector{{Citation needed|date=November 2024}}
<math display="block">|\Psi\rangle =
\sum_{\boldsymbol{\alpha}}\int d^m\!\boldsymbol{\omega}\,\,
\Psi_t(\boldsymbol\alpha,\boldsymbol\omega)\,
|\boldsymbol\alpha,\boldsymbol\omega\rangle</math>
where
* {{math|{{ket|'''α''', '''ω'''}}}} the basis vectors of the chosen representation
* {{math|1=''d<sup>m</sup>'''''ω''' = ''dω''<sub>1</sub>''dω''<sub>2</sub>...''dω<sub>m</sub>''}} a [[differential volume element]] in the continuous degrees of freedom
* <math>\boldsymbol{\Psi}_t(\boldsymbol\alpha, \boldsymbol\omega)</math> a component of the vector <math>|\Psi\rangle</math>, called the '''wave function''' of the system
* {{math|1='''α''' = (''α''<sub>1</sub>, ''α''<sub>2</sub>, ..., ''α<sub>n</sub>'')}} dimensionless discrete quantum numbers
* {{math|1='''ω''' = (''ω''<sub>1</sub>, ''ω''<sub>2</sub>, ..., ''ω<sub>m</sub>'')}} continuous variables (not necessarily dimensionless)

These quantum numbers index the components of the state vector. More, all {{math|'''α'''}} are in an {{math|''n''}}-dimensional [[set (mathematics)|set]] {{math|1=''A'' = ''A''<sub>1</sub> × ''A''<sub>2</sub> × ... × ''A<sub>n</sub>''}} where each {{math|''A<sub>i</sub>''}} is the set of allowed values for {{math|''α<sub>i</sub>''}}; all {{math|'''ω'''}} are in an {{math|''m''}}-dimensional "volume" {{math|Ω ⊆ ℝ<sup>''m''</sup>}} where {{math|1=Ω = Ω<sub>1</sub> × Ω<sub>2</sub> × ... × Ω<sub>''m''</sub>}} and each {{math|Ω<sub>''i''</sub> ⊆ '''R'''}} is the set of allowed values for {{math|''ω<sub>i</sub>''}}, a [[subset]] of the [[real number]]s {{math|'''R'''}}. For generality {{mvar|n}} and {{mvar|m}} are not necessarily equal.

'''Example:'''
{{ ordered list | list-style-type = lower-alpha
| 1 = For a single particle in 3d with spin ''s'', neglecting other degrees of freedom, using Cartesian coordinates, we could take {{math|1='''α''' = (''s<sub>z</sub>'')}} for the spin quantum number of the particle along the z direction, and {{math|1='''ω''' = (''x'', ''y'', ''z'')}} for the particle's position coordinates. Here {{math|1=''A'' = {−''s'', −''s'' + 1, ..., ''s'' − 1, ''s''} }} is the set of allowed spin quantum numbers and {{math|1=Ω = '''R'''<sup>3</sup>}} is the set of all possible particle positions throughout 3d position space.
| 2 = An alternative choice is {{math|1='''α''' = (''s<sub>y</sub>'')}} for the spin quantum number along the y direction and {{math|1='''ω''' = (''p<sub>x</sub>'', ''p<sub>y</sub>'', ''p<sub>z</sub>'')}} for the particle's momentum components. In this case {{math|''A''}} and {{math|Ω}} are the same as before.}}

The [[probability density]] of finding the system at time <math>t</math> at state {{math|{{ket|'''α''', '''ω'''}}}} is
<math display="block">\rho_{\alpha, \omega} (t)= |\Psi(\boldsymbol{\alpha},\boldsymbol{\omega},t)|^2</math>

The probability of finding system with {{math|'''α'''}} in some or all possible discrete-variable configurations, {{math|''D'' ⊆ ''A''}}, and {{math|'''ω'''}} in some or all possible continuous-variable configurations, {{math|''C'' ⊆ Ω}}, is the sum and integral over the density,<ref group="nb">Here: <math display="block">\sum_{\boldsymbol{\alpha}} \equiv \sum_{\alpha_1,\alpha_2,\ldots,\alpha_n} \equiv \sum_{\alpha_1}\sum_{\alpha_2}\cdots\sum_{\alpha_n} </math>is a multiple sum.</ref>
<math display="block">P(t)=\sum_{\boldsymbol{\alpha}\in D}\int_C d^m\!\boldsymbol{\omega}\,\,\rho_{\alpha, \omega}(t)</math>

Since the sum of all probabilities must be 1, the normalization condition
<math display="block">1=\sum_{\boldsymbol{\alpha}\in A}\int_{\Omega}d^m\!\boldsymbol{\omega}\,\,\rho_{\alpha, \omega} (t)</math>
must hold at all times during the evolution of the system.

The normalization condition requires {{math|''ρ d<sup>m</sup>'''''ω'''}} to be dimensionless, by [[dimensional analysis]] {{math|Ψ}} must have the same units as {{math|(''ω''<sub>1</sub>''ω''<sub>2</sub>...''ω<sub>m</sub>'')<sup>−1/2</sup>}}.