Sunday, 16 February 2020

quantum mechanics - Why don't the De Broglie dispersion relation contain a constant term?


Wikipedia says that the dispersion relation for a non-relativistic particle is:



ω=k22m.


But when I tried to calculate it myself, I seem to get a constant term in that formula. My derivation is the following:


Reordering the De Broglie relations I have a generic dispersion relation:


ω=Ekp


Substituting the non-relativistic energy limit:


ω=(mc2+p22m)kp


Substituting the momentum:


ω=(mc2+2k22m)


Performing the division, I get:


ω=mc2+k22m



Maybe I miss something obvious. The relation in the Wikipedia doesn't contain that constant term why? Maybe in the non-relativistic case the mass energy is not considered as energy at all? That would be interesting...



Answer



I believe this is simpler than you make it to be. If you want to substitute in the non-relativistic energy relation, then the correct energy term is just the kinetic energy:


E=p22m


Everything else follows from there:


ω=2k32m×1k=k22m


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