Editing Free particle (section)

===Mathematical description===
{{main|Schrödinger equation|Matter wave}}
A free particle with mass <math>m</math> in non-relativistic quantum mechanics is described by the free [[Schrödinger equation]]:
<math display="block"> - \frac{\hbar^2}{2m} \nabla^2 \ \psi(\mathbf{r}, t) = i\hbar\frac{\partial}{\partial t} \psi (\mathbf{r}, t) </math>

where ''ψ'' is the [[wavefunction]] of the particle at position '''r''' and time ''t''. The solution for a particle with momentum '''p''' or [[wave vector]] '''k''', at [[angular frequency]] ''ω'' or energy ''E'', is given by a [[complex number|complex]] [[plane wave]]:

<math display="block"> \psi(\mathbf{r}, t) = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} = Ae^{i(\mathbf{p}\cdot\mathbf{r} - E t)/\hbar} </math>

with [[amplitude]] ''A'' and has two different rules according to its mass: 
<ol style="list-style-type:lower-alpha;">
<li> if the particle has mass <math>m</math>: <math display="inline">\omega = \frac{\hbar k^2}{2m} </math> (or equivalent <math display="inline">E = \frac{p^2}{2m} </math>). </li>
<li> if the particle is a massless particle: <math>\omega=kc</math>.</li>
</ol>

The eigenvalue spectrum is infinitely degenerate since for each eigenvalue ''E''>0, there corresponds an infinite number of eigenfunctions corresponding to different directions of <math>\mathbf{p}</math>. 

The [[De Broglie relations]]: <math> \mathbf{p} = \hbar \mathbf{k}</math>, <math> E = \hbar \omega</math> apply. Since the potential energy is (stated to be) zero, the total energy ''E'' is equal to the kinetic energy, which has the same form as in classical physics: 

<math display="block"> E = T \,\rightarrow \,\frac{\hbar^2 k^2}{2m} =\hbar \omega </math>

As for ''all'' [[quantum particles]] free ''or'' bound, the [[Heisenberg uncertainty principle]]s <math display="inline"> \Delta p_x \Delta x \geq \frac{\hbar}{2}</math> apply. It is clear that since the plane wave has definite momentum (definite energy), the probability of finding the particle's location is uniform and negligible all over the space. In other words, the wave function is not normalizable in a Euclidean space, ''these stationary states can not correspond to physical realizable states''.<ref>{{Cite web| title=Lecture 9|url=https://mariannasafronova.com/wp-content/uploads/2023/01/424Lecture9.pdf}}</ref>{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|pp=15,19}}