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Dispersion relation for non-relativistic quantum particles

  1. May 26, 2015 #1
    In class I learn that we can get the dispersion relation for particles by using E=hbar*w and p=hbar*k. The calculated phase velocity is w/k = hbar*k/2m, while the group velocity is dw/dk=hbar*k/m. All these make sense to me, except one thing: I always thought that E=hbar*w=hf is only applicable to photons, which are massless. Why can we apply this equation to non-massless particles to obtain such a dispersion relation?
     
  2. jcsd
  3. May 26, 2015 #2
    It sounds like you are talking about phonons. If so then the collective excitation of phonons act as a wave. They are described by wave theory and in particular Bloch Waves, which are to do with the periodicity of the lattice. With regards to your question, since you are treating the particles in a quantum mechanical light, you will inherently be considering wave particle duality or "quantum mechanical behaviour" or however you want to put it. The atoms can be treated as collective phonon or as individual harmonic oscillators.

    If you are learning solid state physics you may recall that when you treat particles as a collective phonon ( i.e with E ~ hf ) you get expressions for, for example heat capacity, which only approach the classical ( 1.5 nR ) in the high T limit. If not look up the Debye Model. Else you can use other methods for the quantum harmonic oscillator approach. So to answer you, it does not only apply to photons. It applies to electrons, nuclear exitations and other systems which classical physics fails.

    I gone off topic of your dispersion a bit sorry, but the ideas are similar. When you have atoms moving in a solid these treatments apply.
     
    Last edited: May 26, 2015
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