Is QM Phase Inherently Relativistic?

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Discussion Overview

The discussion centers around the nature of phase in quantum mechanics (QM) and whether it is inherently relativistic. Participants explore the relationship between phase, momentum, and the relativistic framework, examining implications for both relativistic and non-relativistic quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the phase of a pure momentum state can be expressed as (p.r – Et) / h-bar, suggesting a potential connection to the relativistic action.
  • Another participant mentions that using a relativistic wave equation allows the phase to persist in the non-relativistic limit, though this may not align with the initial inquiry.
  • A reference to Feynman's Lectures is made, where it is explained that the wave function for a particle at rest is constant in space and varies in time, leading to a spatial variation when viewed from a moving frame.
  • One participant reiterates the expression for the phase and questions whether its relationship to the Minkowski space-time dot product is coincidental or indicative of an inherent relativistic nature.
  • Discussion includes the non-relativistic energy expression E=p^2/2m, with a participant noting its operator form in the Schrödinger equation and its connection to the phase expression.

Areas of Agreement / Disagreement

Participants express various viewpoints regarding the relationship between phase and relativity, with no consensus reached on whether phase is inherently relativistic or merely coincidental.

Contextual Notes

Participants reference specific formulations and concepts from quantum mechanics and relativity, indicating a reliance on definitions and interpretations that may not be universally agreed upon.

LarryS
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The phase of a pure momentum state for a single particle traveling freely is

(p.r – Et) / h-bar.

This also happens to be the Minkowski space-time dot product of the 4-momentum and the space-time 4-vector (relativistic action).

Is that just a coincidence or is phase inherently relativistic?
 
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Good question.

If you use a relativistic wave equation this phase somehow survives in the non-relativistic limit. But I guess that is not really the explanation you are expecting.
 
In The Feynman Lectures Vol 3 Ch 7, Feynman begins by postulating that the wave function for a particle at rest is constant in space and varies as [itex]e^{-iE t/ \hbar}[/itex] in time. He then uses the relativistic dot product to derive the wave function for a moving particle.

In other words, a particle at rest can be thought of as being in a plane wave that is headed purely in the time direction. Its wavefronts are planes of constant time. However, with respect to a moving frame, those surfaces of constant time will now be "tilted" in spacetime, leading to the spatial variation of [itex]e^{ipx/\hbar}[/itex].
 
matonski said:
In The Feynman Lectures Vol 3 Ch 7, Feynman begins by postulating that the wave function for a particle at rest is constant in space and varies as [itex]e^{-iE t/ \hbar}[/itex] in time. He then uses the relativistic dot product to derive the wave function for a moving particle.

In other words, a particle at rest can be thought of as being in a plane wave that is headed purely in the time direction. Its wavefronts are planes of constant time. However, with respect to a moving frame, those surfaces of constant time will now be "tilted" in spacetime, leading to the spatial variation of [itex]e^{ipx/\hbar}[/itex].

Interesting. I will have to read up on that.
 
referframe said:
The phase of a pure momentum state for a single particle traveling freely is

(p.r – Et) / h-bar.

This also happens to be the Minkowski space-time dot product of the 4-momentum and the space-time 4-vector (relativistic action).

Is that just a coincidence or is phase inherently relativistic?

In non-relativistic QM, E=p^2/2m.
 
Meir Achuz said:
In non-relativistic QM, E=p^2/2m.

Yes, and that expression normally occurs in operator form on left side of the Schrödinger equation. But the right side of that equation and also the definition of the momentum and energy operators is based on the expression (p.r - Et)/h-bar .
 

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