Are De Broglie relations only applicable to particles that have zero potential energy?De Broglie relations are always written as: E = h ν p = h λ However, it doesn't make sense when you have waves that are eigenstates of a Hamiltonian operator with a non-zero potential. That is because that would give us a direct relation between momentum and energy: E p = v p , where v p is the phase velocity. This doesn't seem to make sense to me. So, would these equations be the same with the electron of the hydrogen atom, for example?

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Are De Broglie relations only applicable to particles that have zero potential energy?De Broglie relations are always written as:  E  =  h  ν  p  =      h    λ  However, it doesn&#039;t make sense when you have waves that are eigenstates of a Hamiltonian operator with a non-zero potential. That is because that would give us a direct relation between momentum and energy:       E    p    =      v    p  , where       v    p   is the phase velocity. This doesn&#039;t seem to make sense to me. So, would these equations be the same with the electron of the hydrogen atom, for example?

Kaylin Barry · Accepted Answer

De Broglie&#039;s proposition is a limited-validity rule assigning a travelling wave to a freely moving particle. If we have eigenstates of a Hamiltonian with some potential term, we refer to Schroedinger&#039;s theory instead, which describes more complicated situations than de Broglie relations do.The relation  ν  =      E    h  is still valid in Schroedinger&#039;s description of Hamiltonian eigenstates, here   ν is frequency of oscillation of function   Ψ. This reminds of de Broglie&#039;s wave, but   Ψ of an Hamiltonian eigenstate is usually not a plane wave. It is usually some function concentrated in those regions of configuration space that are probable configurations of the system. For single electron atom,   Ψ is concentrated near the nucleus and it is not a travelling wave.Hence, the relation  λ  =      h    p  is usually no longer applicable in this context, as there is no single momentum   p of the electron. Energy eigenstates have definite energy, but usually do not have definite momentum.

Are De Broglie relations only applicable to particles that have zero potential energy? De Broglie

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