Editing Wave function (section)

=== 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.