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With the formation of Einstein special relativity, $E^2 = p^2c^2 + m^2c^4$, could the term of mass be replaced with the De Broglie equation of material waves while $p$, momentum, be replaced with the momentum of the wave as $hk$ where $k$ is the wavenumbers?

B. D.
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    https://en.wikipedia.org/wiki/Klein-Gordon_equation#Derivation – J.G. May 21 '18 at 21:02
  • Thank you for the respond, I know that; but if the momentum could be written as hk, and mass could be written as h/v(lambda), will it be the energy of the waves from the special relativity? – B. D. May 21 '18 at 22:27
  • $p=\hbar k$ is same as de Broglie relation, $$p=\frac{h}{\lambda}$$ since $k=\frac{2 \pi}{\lambda}$ and $\hbar = 2\pi/h $. Probably you meant Compton wavelength, $\lambda_C = \frac{h}{mc}$?? If so, do not forget that Compton wavelength is not the wavelength of the particle, it is instead the wavelength of the photon that scatters from it due to its mass. – Oktay Doğangün May 21 '18 at 22:53
  • I meant matter-wave directly, not Compton wavelength. If I use hk to replace the momentum for waves, h/v(lambda) to replace the mass; the special relativity will be $E^2 = h^2k^2c^2 + (h/v(lambda))^2c^4$. What would this energy mean? Moreover, by simple substitution, this term of energy will contain the energy of photon from photoelectric effect as well. – B. D. May 21 '18 at 22:57
  • "[C]ould the term of mass be replaced with the De Broglie equation of material waves[?]" -- How do you replace a real number (i.e., mass) with an equation (i.e., the De Broglie equation)? It is unclear as to what you mean. Voting to close. –  May 15 '20 at 02:24
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    I think this might help you somewhat. – PNS May 15 '20 at 07:49
  • Related : About de Broglie relations, what exactly is E ? Its energy of what?. – Frobenius Oct 25 '20 at 13:20

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You can't do that because it's the wrong "mass". The $m$ in the formula for $E^2$ is rest mass, not relativistic mass.

J.G.
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  • You meant the m in the special relativity was the rest mass? But how about that momentum term, $ E = mc^2$ was for the rest mass. But when De Broglie used $ E = mc^2$ for the rest mass and $ E = hv$ for photoelectric effect, they could work. – B. D. May 22 '18 at 05:22
  • @B.D. No, $E=mc^2$ uses the relativistic mass, which is why it's consistent with the rest-mass-using equation you quoted. – J.G. May 22 '18 at 05:43
  • Since the De Broglie equation was from the process of derivation of special relativity and photoelectric effect equation, will the energy of the matter wave from https://en.wikipedia.org/wiki/Matter_wave be compatible with the energy term from the special relativity with the momentum term $ p^2c^2$ ? – B. D. May 22 '18 at 16:41
  • @B.D. You need $E=h\nu=\hbar\omega$. – J.G. May 22 '18 at 16:53
  • @ J.G. could $ E = hv $ be compatible for the energy waves? Wasn't it only served for the photon energy? Could $ E = hv $ work for the matter wave as well? – B. D. May 22 '18 at 16:54
  • @B.D. Yes, it would. – J.G. May 22 '18 at 17:00
  • since it would work for the matter wave, could it, $ E=hv $ be used as an energy operator for the hamiltonian as well? – B. D. May 22 '18 at 17:05