De Broglie Wavelength - How does this hold for slow objects?

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Discussion Overview

The discussion revolves around the application of the de Broglie wavelength equation, specifically 'lambda = h / p', in the context of slow or non-relativistic objects. Participants explore the implications of energy and momentum relationships in both relativistic and non-relativistic regimes, raising questions about the validity of the equation under different conditions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the applicability of the de Broglie wavelength equation when the rest mass energy (mc^2) is comparable to the momentum energy (cp).
  • Others argue that the relationships between energy, momentum, frequency, and wavelength hold generally, and that the transition between non-relativistic and relativistic limits is crucial for understanding the equation's application.
  • A participant suggests that the de Broglie relation may only apply strictly to relativistic particles, as this is when mc^2 can be neglected.
  • Another viewpoint is presented that the de Broglie wavelength can apply to both relativistic and non-relativistic conditions, emphasizing that the mass used in calculations is the rest mass and that relativistic corrections must be considered.
  • Discussion includes the historical context of de Broglie's waves compared to the quantum mechanical wave function, with some participants noting that de Broglie's waves may be interpreted as faster than light, raising further questions about their implications in quantum mechanics.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the de Broglie wavelength equation is applicable in non-relativistic conditions, with multiple competing views remaining regarding the treatment of rest mass energy and the nature of de Broglie's waves versus the quantum wave function.

Contextual Notes

Limitations in the discussion include assumptions about the neglect of rest mass energy in certain conditions, the dependence on definitions of momentum and energy, and the unresolved nature of the relationship between de Broglie's waves and the quantum wave function.

sam_p_r
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OK so you get to the matter wave equation 'lambda = h / p' using E=cp - which describes the energy for massless particles. I can understand this holding for when cp>>mc^2 , but not for when the mc^2 is comparible. Any help?
 
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Well the relations between energy/momentum and frequency/wavelength hold generally:
E=h\nu, p=h/\lambda. It's the relation between E and p which is relevant in going from non-relativistic to relativistic to ultra-relativistic limits (as determined by the ratio pc/mc^{2}). You could always use E^{2}=(pc)^{2}+(mc^{2})^{2}. We ignore the second term in the ultra-relativistic limit, while in the non-relativistic limit, it simplifies to E=p^{2}/2m. Otherwise, use the full expression.
 
Last edited:
javierR said:
Well the relations between energy/momentum and frequency/wavelength hold generally:
E=h\nu, p=h/\lambda. It's the relation between E and p which is relevant in going from non-relativistic to relativistic to ultra-relativistic limits (as determined by the ratio pc/mc^{2}). You could always use E^{2}=(pc)^{2}+(mc^{2})^{2}. We ignore the second term in the ultra-relativistic limit, while in the non-relativistic limit, it simplifies to E=p^{2}/2m. Otherwise, use the full expression.

Could you elaborate a little more about the ultra-relativistic case?
 
sam_p_r said:
OK so you get to the matter wave equation 'lambda = h / p' using E=cp - which describes the energy for massless particles. I can understand this holding for when cp>>mc^2 , but not for when the mc^2 is comparible. Any help?

What specific problem do you see?.
 
alexepascual said:
What specific problem do you see?.

That E does not equal cp, it equals the square root of (cp)^2 + (mc^2)^2. So for de broglie equation, you have to neglect mc^2, which I can see as reasonable if it is very small in comparison to cp, but there are many occasions where it won't be.

de broglie combines E=hc/lambda and E=cp,

So if you don't have E=cp you can't get the de broglie relation as it is.
 
javierR said:
Well the relations between energy/momentum and frequency/wavelength hold generally:
E=h\nu, p=h/\lambda. It's the relation between E and p which is relevant in going from non-relativistic to relativistic to ultra-relativistic limits (as determined by the ratio pc/mc^{2}). You could always use E^{2}=(pc)^{2}+(mc^{2})^{2}. We ignore the second term in the ultra-relativistic limit, while in the non-relativistic limit, it simplifies to E=p^{2}/2m. Otherwise, use the full expression.

So what you're saying is that it only applies (in the exact lambda=h/p) for relativistic particles? (as its the only time where you can neglect mc^2).

Because in many examples in textbooks it applies the formula to non-relativistic conditions.
 
sam_p_r said:
So what you're saying is that it only applies (in the exact lambda=h/p) for relativistic particles? (as its the only time where you can neglect mc^2).

Because in many examples in textbooks it applies the formula to non-relativistic conditions.

I think it applies to both relativistic and non-relativistic conditions. The m that you list is actually m0, the rest mass. P should be the relativistic momentum and E is the relativistic energy. These both have a correction due to speed of the particle (or refference frame).
In the limit where v=c this correction disappears.
I don't think the rest mass energy can be neglected in either the relativisti or non-relativistic case. But I may be wrong.
I think the de Broglie waves have been superceeded by the QM wave funtion, but I haven't looked at the difference. I know that the Schrödinger equation is not Lorentz-invariant and that the Dirac equation is. I would assume that there is no problem with the wave function itself. It would be nice to compare the de Broglie wave with the wave function. This must be done in many books but I can't find it in the ones I have. Maybe I can find something on the Web.
 
It appears that there is no difference between de Broglie's waves and the wave function. I looks like de Broglie calculated things like phase velocity and wave length but he didn't come up with an expression for the wave and that Scrodinger did that. So, it would appear that de Broglie's waves and the wave function refer to the same waves.
Now, reading again about de Broglie's waves, these are faster than light. I don't remember any mention in QM about the wave funtion of a free particle being faster than light. Maybe I just didn't pay attention to that.
Maybe that's where the energy corresponding to the rest mass comes into play!

If anybody can make this clear for us I'll appreciate it.
 

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