QM Phase & Four-Vectors

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  • #1
referframe
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Main Question or Discussion Point

The QM phase of a single particle traveling freely in 3 dimensions is (rp – Et), where r and p are the 3-D position and momentum vectors. This is also the dot product of the space-time four-vector (r,-t) with the Energy-Momentum four-vector (p,E)

Is this "coincidence" related to relativistic QM?

Thanks in advance.
 
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Answers and Replies

  • #2
clem
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No. It is not a coincidence. It is the case in any relativistic wave, including classical EM.
 
  • #3
referframe
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No. It is not a coincidence. It is the case in any relativistic wave, including classical EM.
What classical EM equation(s) contain this term? I looked at the original version of Maxwell's Equations and could not see any connection. Thanks.
 
  • #4
clem
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A classical EM wave is [tex]{\bf E}\exp[i({\bf k\cdot r}-\omega t)][/tex].
 
  • #5
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It is a consequence of the fact that relativistic fields satisfies the relativistic wave equation which in the non-relativistic limit gives the Schrodinger eq, and of the de Broglie relation: E = \hbar \omega, p = \hbar k. As a consequence the phase of a field is Lorentz invariant and the four-vector (\omega, k) is covariant.
 

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