Editing Four-vector (section)

==Waves==

===Four-frequency===

A photonic [[plane wave]] can be described by the ''[[four-frequency]]'', defined as

<math display="block">\mathbf{N} = \nu\left(1 , \hat{\mathbf{n}} \right)</math>

where {{mvar|ν}} is the frequency of the wave and <math>\hat{\mathbf{n}}</math> is a [[unit vector]] in the travel direction of the wave. Now:

<math display="block">\|\mathbf{N}\| = N^\mu N_\mu = \nu ^2 \left(1 - \hat{\mathbf{n}}\cdot\hat{\mathbf{n}}\right) = 0</math>

so the four-frequency of a photon is always a null vector.

===Four-wavevector===

{{see also|De Broglie relation}}

The quantities reciprocal to time {{mvar|t}} and space '''{{math|r}}''' are the [[angular frequency]] {{mvar|ω}} and [[angular wave vector]] '''{{math|k}}''', respectively. They form the components of the '''four-wavevector''' or '''wave four-vector''':

<math display="block">\mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \left(\frac{\omega}{c}, \frac{\omega}{v_p} \hat\mathbf{n}\right) \,.</math>

The wave four-vector has [[coherent derived unit]] of [[reciprocal meters]] in the SI.<ref name="o144">{{cite web | title=Details for IEV number 113-07-57: "four-wave vector" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=113-07-57 | language=ja | access-date=2024-09-08}}</ref>

A wave packet of nearly [[monochromatic]] light can be described by:

<math display="block">\mathbf{K} = \frac{2\pi}{c}\mathbf{N} = \frac{2\pi}{c} \nu\left(1,\hat{\mathbf{n}}\right) = \frac{\omega}{c} \left(1, \hat{\mathbf{n}}\right) ~.</math>

The de Broglie relations then showed that four-wavevector applied to [[matter wave]]s as well as to light waves:
<math display="block">\mathbf{P} = \hbar \mathbf{K} = \left(\frac{E}{c},\vec{p}\right) = \hbar \left(\frac{\omega}{c},\vec{k} \right) ~.</math>
yielding <math>E = \hbar \omega</math> and <math>\vec{p} = \hbar \vec{k}</math>, where {{mvar|ħ}} is the [[Planck constant]] divided by {{math|2''π''}}&nbsp;.

The square of the norm is:
<math display="block">\| \mathbf{K} \|^2 = K^\mu K_\mu = \left(\frac{\omega}{c}\right)^2 - \mathbf{k}\cdot\mathbf{k} \,,</math>
and by the de Broglie relation:
<math display="block">\| \mathbf{K} \|^2 = \frac{1}{\hbar^2} \| \mathbf{P} \|^2 = \left(\frac{m_0 c}{\hbar}\right)^2 \,,</math>
we have the matter wave analogue of the energy–momentum relation:
<math display="block">\left(\frac{\omega}{c}\right)^2 - \mathbf{k}\cdot\mathbf{k} = \left(\frac{m_0 c}{\hbar}\right)^2 ~.</math>

Note that for massless particles, in which case {{math|''m''<sub>0</sub> {{=}} 0}}, we have:
<math display="block">\left(\frac{\omega}{c}\right)^2 = \mathbf{k}\cdot\mathbf{k} \,,</math>
or {{math|‖'''k'''‖ {{=}} ''ω''/''c''}}&nbsp;. Note this is consistent with the above case; for photons with a 3-wavevector of modulus {{nobr|{{math|''ω / c''}} ,}} in the direction of wave propagation defined by the unit vector <math>\ \hat{\mathbf{n}} ~.</math>