Editing Free particle (section)

==Quantum free particle==

[[File:Propagation of a de broglie wave.svg|290px|right|thumb|Propagation of [[matter wave|de Broglie waves]] in 1d - real part of the [[complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the colour [[opacity (optics)|opacity]]) of finding the particle at a given point ''x'' is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the [[curvature]] decreases, so the decreases again, and vice versa - the result is an alternating amplitude: a wave. Top: [[Plane wave]]. Bottom: [[Wave packet]].]]

===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}}

===Measurement and calculations===
The normalization condition for the wave function states that if a wavefunction belongs to the [[quantum state space]]{{sfn|Blanchard|Brüning|2015|p=210}} 
<math display="block">\psi \in L^2(\mathbb{R}^3),</math>
then the integral of the [[probability density function]]
<math display="block"> \rho(\mathbf{r},t) = \psi^*(\mathbf{r},t)\psi(\mathbf{r},t) = |\psi(\mathbf{r},t)|^2,</math>

where * denotes [[complex conjugate]], over all space is the probability of finding the particle in all space, which must be unity if the particle exists:
<math display="block"> \int_{\mathbb{R}^3} |\psi(\mathbf{r},t)|^2 d^3 \mathbf{r}=1.</math>
The state of a free particle given by plane wave solutions is ''not'' normalizable as
<math display="block">Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} \notin L^{2}(\mathbb{R}^3),</math>
for any fixed time <math>t</math>. Using [[wave packet]]s, however, the states can be expressed as functions that ''are'' normalizable.

{{Clear}}
{{multiple image
 | align = center
 | direction = horizontal
 | footer    = Interpretation of wave function for one spin-0 particle in one dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (which can measure in %) of finding the particle at the points on the x-axis.
 | image1    = Quantum mechanics travelling wavefunctions.svg
 | caption1  = Increasing amounts of wavepacket localization, meaning the particle becomes more localized.
 | width1    = 400
 | image2    = Perfect localization.svg
 | caption2  = In the limit ''ħ'' → 0, the particle's position and momentum become known exactly.
 | width2    = 200
}}
{{Clear}}

===Wave packet===
{{main|Wave packet}}
Using the [[Fourier inversion theorem]], the free particle wave function may be represented by a superposition of ''momentum'' eigenfunctions, or, ''wave packet'':<ref>{{harvnb|Hall|2013}} Section 4.1</ref>
<math display="block"> \psi(\mathbf{r}, t) =\frac{1}{(\sqrt{2\pi})^3} \int_\mathrm{all \, \mathbf{k} \, space} \hat \psi_0 (\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega(\mathbf{k}) t)} d^3 \mathbf{k},</math>
where 
<math display="block"> \omega(\mathbf{k}) = \frac{\hbar \mathbf{k}^2}{2m},</math>
and <math>\hat \psi_0 (\mathbf{k})</math> is the [[Fourier transform]] of a "[[Fourier_inversion_theorem#Conditions_on_the_function|sufficiently nice]]" initial wavefunction <math>\psi(\mathbf{r},0)</math>. 

The expectation value of the momentum '''p''' for the complex plane wave is

<math display="block"> \langle\mathbf{p}\rangle=\left\langle \psi \left|-i\hbar\nabla\right|\psi\right\rangle = \hbar\mathbf{k} ,</math>

and for the general wave packet it is

<math display="block"> \langle\mathbf{p}\rangle = \int_\mathrm{all\,space} \psi^*(\mathbf{r},t)(-i\hbar\nabla)\psi(\mathbf{r},t) d^3 \mathbf{r} = \int_\mathrm{all \, \textbf{k} \, space} \hbar \mathbf{k} |\hat\psi_0(\mathbf{k})|^2 d^3 \mathbf{k}. </math>

The expectation value of the energy E is 

<math display="block"> \langle E\rangle=\left\langle \psi \left|- \frac{\hbar^2}{2m} \nabla^2 \right|\psi\right\rangle = \int_\text{all space} \psi^*(\mathbf{r},t)\left(- \frac{\hbar^2}{2m} \nabla^2 \right)\psi(\mathbf{r},t) d^3 \mathbf{r} .</math>

===Group velocity and phase velocity===
[[File:Wave_packet_propagation.png|thumb|right|Propagation of a wave packet, with the motion of a single peak shaded in purple. The peaks move at the phase velocity while the overall packet moves at the group velocity.]]
The [[phase velocity]] is defined to be the speed at which a plane wave solution propagates, namely

<math display="block"> v_p=\frac{\omega}{k}=\frac{\hbar k}{2m} = \frac{p}{2m}. </math>

Note that <math>\frac{p}{2m}</math> is ''not'' the speed of a classical particle with momentum <math>p</math>; rather, it is half of the classical velocity.

Meanwhile, suppose that the initial wave function <math>\psi_0</math> is a [[wave packet]] whose Fourier transform <math>\hat\psi_0</math> is concentrated near a particular wave vector <math>\mathbf k</math>. Then the [[group velocity]] of the plane wave is defined as
<math display="block"> v_g= \nabla\omega(\mathbf k)=\frac{\hbar\mathbf k}{m}=\frac{\mathbf p}{m},</math>

which agrees with the formula for the classical velocity of the particle. The group velocity is the (approximate) speed at which the whole wave packet propagates, while the phase velocity is the speed at which the individual peaks in the wave packet move.<ref>{{harvnb|Hall|2013}} Sections 4.3 and 4.4</ref> The figure illustrates this phenomenon, with the individual peaks within the wave packet propagating at half the speed of the overall packet.

===Spread of the wave packet===
The notion of group velocity is based on a linear approximation to the dispersion relation <math>\omega(k)</math> near a particular value of <math>k</math>.<ref>{{harvnb|Hall|2013}} Equation 4.24</ref> In this approximation, the amplitude of the wave packet moves at a velocity equal to the group velocity ''without changing shape''. This result is an approximation that fails to capture certain interesting aspects of the evolution a free quantum particle. Notably, the width of the wave packet, as measured by the uncertainty in the position, grows linearly in time for large times. This phenomenon is called the [[Wave_packet#Gaussian_wave_packets_in_quantum_mechanics |spread of the wave packet]] for a free particle.

Specifically, it is not difficult to compute an exact formula for the uncertainty <math>\Delta_{\psi(t)}X</math> as a function of time, where <math>X</math> is the position operator. Working in one spatial dimension for simplicity, we have:<ref>{{harvnb|Hall|2013}} Proposition 4.10</ref>
<math display="block">(\Delta_{\psi(t)}X)^2 = \frac{t^2}{m^2}(\Delta_{\psi_0}P)^2+\frac{2t}{m}\left(\left\langle \tfrac{1}{2}({XP+PX})\right\rangle_{\psi_0} - \left\langle X\right\rangle_{\psi_0} \left\langle P\right\rangle_{\psi_0} \right)+(\Delta_{\psi_0}X)^2,</math>
where <math>\psi_0</math> is the time-zero wave function. The expression in parentheses in the second term on the right-hand side is the quantum covariance of <math>X</math> and <math>P</math>.

Thus, for large positive times, the uncertainty in <math>X</math> grows linearly, with the coefficient of <math>t</math> equal to <math>(\Delta_{\psi_0}P)/m</math>. If the momentum of the initial wave function <math>\psi_0</math> is highly localized, the wave packet will spread slowly and the group-velocity approximation will remain good for a long time. Intuitively, this result says that if the initial wave function has a very sharply defined momentum, then the particle has a sharply defined velocity and will (to good approximation) propagate at this velocity for a long time.