Understanding QM Phase & Four-Vectors: Exploring Relativity in Quantum Mechanics

[LarryS]

The QM phase of a single particle traveling freely in 3 dimensions is (r•p – 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.

 

[Meir Achuz]

No. It is not a coincidence. It is the case in any relativistic wave, including classical EM.

 

[LarryS]

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.

 

[Meir Achuz]

A classical EM wave is $${\bf E}\exp[i({\bf k\cdot r}-\omega t)]$$.

 

[naturale]

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.