What is the relationship between rest energy and frequency in quantum mechanics?

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Discussion Overview

The discussion revolves around the relationship between rest energy and frequency in quantum mechanics, particularly focusing on how the energy associated with rest mass fits into the framework of particle frequency. Participants explore concepts related to intrinsic frequency, phase velocity, and the implications of rest mass on these properties.

Discussion Character

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

Main Points Raised

  • Some participants assert that the energy of a particle with non-zero rest mass is given by Planck's constant times its frequency, raising questions about how rest mass energy (m0c²) integrates into this model.
  • One participant references Louis de Broglie's proposal that frequency can be expressed in terms of mass, specifically \(\nu = \frac{mc^2}{h}\), and discusses the derivation involving relativistic factors.
  • Another participant expresses confusion regarding the implications of a particle at rest having a non-zero frequency and wavelength, suggesting it possesses a phase velocity even when stationary.
  • There is a mention of the phase velocity of a massive particle exceeding the speed of light, with a note that this does not violate special relativity since it does not convey information.
  • Participants contrast phase velocity with group velocity and particle velocity, noting that the latter must be less than c for massive particles.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and confusion regarding the implications of rest mass on frequency and phase velocity. There is no consensus on the interpretation of these concepts, and multiple viewpoints are presented without resolution.

Contextual Notes

The discussion includes references to historical contributions and definitions that may not be universally accepted or clarified, leaving some assumptions and interpretations open to further exploration.

LarryS
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In QM, the energy of a particle with a non-zero rest mass is Planck's constant times the frequency. How does the energy associated with the rest mass m0c2 fit into this picture? Planck wrote of an "instrinsic frequency" associated with the rest mass. Thanks in advance.
 
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referframe said:
In QM, the energy of a particle with a non-zero rest mass is Planck's constant times the frequency. How does the energy associated with the rest mass m0c2 fit into this picture?...

Well, setting
[tex]mc^2 = h\nu,[/tex]
we can express the frequency in terms of the mass:
[tex]\nu = \frac{mc^2}{h},[/tex]
which was first proposed by Louis de Broglie in 1924. Actually, de Broglie derived
[tex]\nu = \frac{\gamma mc^2}{h} = \frac{mc^2}{h\sqrt{1 - \frac{v^2}{c^2}}},[/tex]
which reduces to the previous expression for [tex]v = 0[/tex]. More information at Wikipedia.
 
daschaich said:
Well, setting
[tex]mc^2 = h\nu,[/tex]
we can express the frequency in terms of the mass:
[tex]\nu = \frac{mc^2}{h},[/tex]
which was first proposed by Louis de Broglie in 1924. Actually, de Broglie derived
[tex]\nu = \frac{\gamma mc^2}{h} = \frac{mc^2}{h\sqrt{1 - \frac{v^2}{c^2}}},[/tex]
which reduces to the previous expression for [tex]v = 0[/tex]. More information at Wikipedia.

I guess I'm still a little bit confused. So, a particle with a rest mass (m0>0) which is at rest (v = 0) has a non-zero frequency and a non-zero wave length and therefore has a phase velocity while the particle is at rest...?
 
referframe said:
I guess I'm still a little bit confused. So, a particle with a rest mass (m0>0) which is at rest (v = 0) has a non-zero frequency and a non-zero wave length and therefore has a phase velocity while the particle is at rest...?

Sounds like it. In fact, the Wikipedia article I mentioned above links to this one, where we learn that the phase velocity of a massive particle always exceeds the speed of light c: "The superluminal phase velocity does not violate special relativity, for it doesn't carry any information."

The article goes on to contrast the phase velocity with the group velocity and the particle velocity. The particle velocity of a massive particle must be less than c, and is what vanishes for a particle at rest.
 

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