Editing Matter wave (section)

== Traveling matter waves <span class="anchor" id="De Broglie relations"></span> ==
<!--This section is linked to by [[de Broglie relations]] -->
Waves have more complicated concepts for [[Group velocity|velocity]] than solid objects.
The simplest approach is to focus on the description in terms of plane matter waves for a [[free particle]], that is a wave function described by
<math display="block">\psi (\mathbf{r}) = e^{ i \mathbf{k} \cdot \mathbf{r}-i \omega t },</math>
where <math>\mathbf{r}</math> is a position in real space, <math>\mathbf{k}</math>  is the [[wavevector|wave vector]] in units of inverse meters, {{math|''ω''}} is the [[angular frequency]] with units of inverse time and <math>t</math> is time. (Here the physics definition for the wave vector is used, which is <math>2 \pi</math> times the wave vector used in [[crystallography]], see [[wavevector]].) The de Broglie equations relate the [[wavelength]] {{math|''λ''}} to the modulus of the [[momentum]] <math>|\mathbf{p}| = p</math>, and [[frequency]] {{math|''f''}} to the total energy {{math|''E''}} of a [[free particle]] as written above:<ref name="Resnick 1985">{{cite book |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles |edition=2nd |first1=R. |last1=Resnick |first2=R. |last2=Eisberg |publisher=John Wiley & Sons |date=1985 |location=New York |isbn=978-0-471-87373-0 |url=https://archive.org/details/quantumphysicsof00eisb }}</ref>
<math display="block">\begin{align}
& \lambda = \frac {2 \pi}{|\mathbf{k}|} = \frac{h}{p}\\
& f = \frac{\omega}{2 \pi}= \frac{E}{h}
\end{align}</math>
where {{math|''h''}} is the [[Planck constant]]. The equations can also be written as
<math display="block">\begin{align}
& \mathbf{p} = \hbar \mathbf{k}\\
& E = \hbar \omega ,\\
\end{align}</math>
Here, {{math|1=''ħ'' = ''h''/2''π''}} is the reduced Planck constant. The second equation is also referred to as the [[Planck–Einstein relation]].

=== Group velocity ===
In the de Broglie hypothesis, the velocity of a particle equals the [[group velocity]] of the matter wave.<ref name="WhittakerII" />{{rp|214}}
In isotropic media or a vacuum the [[group velocity]] of a wave is defined by:
<math display="block"> \mathbf{v_g} = \frac{\partial \omega(\mathbf{k})}{\partial \mathbf{k}} </math>
The relationship between the angular frequency and wavevector is called the [[Dispersion relation#De Broglie dispersion relations|dispersion relationship]]. For the non-relativistic case this is:
<math display="block">\omega(\mathbf{k}) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.</math>
where <math>m_0</math> is the rest mass. Applying the derivative gives the (non-relativistic) '''matter wave group velocity''':
<math display="block">\mathbf{v_g} = \frac{\hbar \mathbf{k}}{m_0}</math>
For comparison, the group velocity of light, with a [[Dispersion_relation#Electromagnetic_waves_in_vacuum|dispersion]] <math>\omega(k)=ck</math>, is the [[speed of light]] <math>c</math>.

As an alternative, using the relativistic [[Dispersion relation#De Broglie dispersion relations|dispersion relationship]] for matter waves
<math display="block"> \omega(\mathbf{k}) = \sqrt{k^2c^2 + \left(\frac{m_0c^2}{\hbar}\right)^2} \,.</math>
then
<math display="block">\mathbf{v_g} = \frac{\mathbf{k}c^2}{\omega} </math>
This relativistic form relates to the phase velocity as discussed below.

For non-isotropic media we use the [[Energy–momentum relation|Energy–momentum]] form instead:
<math display="block">\begin{align}
  \mathbf{v}_\mathrm{g} &= \frac{\partial \omega}{\partial \mathbf{k}} = \frac{\partial (E/\hbar)}{\partial (\mathbf{p} /\hbar)} = \frac{\partial E}{\partial \mathbf{p}} = \frac{\partial}{\partial \mathbf{p}} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\
    &= \frac{\mathbf{p} c^2}{\sqrt{p^2c^2 + m_0^2c^4}}\\
    &= \frac{\mathbf{p} c^2}{E} .
\end{align}</math>

But (see below), since the phase velocity is <math>\mathbf{v}_\mathrm{p} = E/\mathbf{p} = c^2/\mathbf{v}</math>, then
<math display="block">\begin{align}
 \mathbf{v}_\mathrm{g} &= \frac{\mathbf{p}c^2}{E}\\
    &= \frac{c^2}{\mathbf{v}_\mathrm{p}}\\
    &= \mathbf{v} ,
\end{align}</math>
where <math>\mathbf{v}</math> is the velocity of the center of mass of the particle, identical to the group velocity.

=== Phase velocity ===
The [[phase velocity]] in isotropic media is defined as:
<math display="block">\mathbf{v_p} = \frac{\omega}{\mathbf{k}}</math>
Using the relativistic group velocity above:<ref name="WhittakerII"/>{{rp|p=215}}
<math display="block">\mathbf{v_p} = \frac{c^2 }{\mathbf{v_g}}</math>
This shows that <math>\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2</math> as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey  <math>\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2</math>, as both <math>|\mathbf{v_p}|=c</math> and <math>|\mathbf{v_g}|=c</math>. Since for matter waves, <math>|\mathbf{v_g}| < c</math>, it follows that <math>|\mathbf{v_p}| > c</math>, but only the group velocity carries information. The [[Faster-than-light|superluminal]] phase velocity therefore does not violate special relativity, as it does not carry information.

For non-isotropic media, then
<math display="block">\mathbf{v}_\mathrm{p} = \frac{\omega}{\mathbf{k}} = \frac{E/\hbar}{\mathbf{p}/\hbar} = \frac{E}{\mathbf{p}}. </math>

Using the [[special relativity|relativistic]] relations for energy and momentum yields
<math display="block">\mathbf{v}_\mathrm{p} = \frac{E}{\mathbf{p}} = \frac{m c^2}{m \mathbf{v}} = \frac{\gamma m_0 c^2}{\gamma m_0 \mathbf{v}} = \frac{c^2}{\mathbf{v}}.</math>
The variable <math>\mathbf{v}</math> can either be interpreted as the speed of the particle or the group velocity of the corresponding matter wave—the two are the same. Since the particle speed <math>|\mathbf{v}| < c </math> for any particle that has nonzero mass (according to [[special relativity]]), the phase velocity of matter waves always exceeds ''c'', i.e.,
<math display="block">| \mathbf{v}_\mathrm{p} | > c ,</math>
which approaches ''c'' when the particle speed is relativistic. The [[Faster-than-light|superluminal]] phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article on ''[[Dispersion (optics)#Group velocity dispersion|Dispersion (optics)]]'' for further details.

=== Special relativity ===
Using two formulas from [[special relativity]], one for the relativistic mass energy and one for the [[Momentum#Relativistic|relativistic momentum]]
<math display="block">\begin{align}
E &= m c^2 = \gamma m_0 c^2 \\[1ex]
\mathbf{p} &= m\mathbf{v} = \gamma m_0 \mathbf{v}
\end{align} </math>
allows the equations for de Broglie wavelength and frequency to be written as
<math display="block">\begin{align}
&\lambda =\,\, \frac {h}{\gamma m_0v}\, =\, \frac {h}{m_0v}\,\,\, \sqrt{1 - \frac{v^2}{c^2}} \\[2.38ex]
& f = \frac{\gamma\,m_0 c^2}{h} = \frac {m_0 c^2}{h\sqrt{1 - \frac{v^2}{c^2}}} ,
\end{align}</math>
where <math>v=|\mathbf{v}|</math> is the [[velocity]], <math>\gamma</math> the [[Lorentz factor]], and <math>c</math> the [[speed of light]] in vacuum.<ref>{{cite book |title=Stationary states |first=Alan |last=Holden |publisher=Oxford University Press |date=1971 |location=New York |isbn=978-0-19-501497-6 }}</ref><ref>Williams, W.S.C. (2002). ''Introducing Special Relativity'', Taylor & Francis, London, {{ISBN|0-415-27761-2}}, p. 192.</ref> This shows that as the velocity of a particle approaches zero (rest) the de Broglie wavelength approaches infinity.

=== Four-vectors ===
{{Main|Four-vector}}

Using four-vectors, the de Broglie relations form a single equation:
<math display="block">\mathbf{P}= \hbar\mathbf{K} ,</math>
which is [[Inertial frame of reference|frame]]-independent.
Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:
<math display="block">\mathbf{K} = \left(\frac{\omega_0}{c^2}\right)\mathbf{U} ,</math>
where
* [[Four-momentum]] <math>\mathbf{P} = \left(\frac{E}{c}, {\mathbf{p}} \right)</math>
* [[Four-vector#Four-wavevector|Four-wavevector]] <math>\mathbf{K} = \left(\frac{\omega}{c}, {\mathbf{k}} \right) </math>
* [[Four-velocity]] <math>\mathbf{U} = \gamma(c,{\mathbf{u}}) = \gamma(c,v_\mathrm{g} \hat{\mathbf{u}}) </math>