# Dispersion relation for non-relativistic quantum particles

• dilloncyh
In summary, in class it was discussed that the dispersion relation for particles can be obtained using the equations E=hbar*w and p=hbar*k. The resulting phase and group velocities make sense, but there is confusion about applying E=hbar*w to non-massless particles. This can be explained by considering wave-particle duality and treating the particles as collective phonons. This approach is used in solid state physics, where the Debye Model is used to calculate properties such as heat capacity. This concept can also be applied to other systems besides photons, such as electrons and nuclear excitations.
dilloncyh
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?

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.

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## 1. What is a dispersion relation?

A dispersion relation is a mathematical relationship that describes how the energy of a particle depends on its momentum. In other words, it shows how the energy of a particle changes as its momentum changes.

## 2. What is the difference between relativistic and non-relativistic dispersion relations?

In a non-relativistic dispersion relation, the energy of a particle is proportional to its momentum squared, while in a relativistic dispersion relation, the energy is proportional to the square root of its momentum squared. This is due to the effects of special relativity, which become significant at high speeds.

## 3. How is the dispersion relation used in quantum mechanics?

The dispersion relation is an important tool in quantum mechanics because it allows us to calculate the energy of a particle based on its momentum. This is especially useful in understanding the behavior of particles at the atomic and subatomic level.

## 4. How does the dispersion relation relate to the uncertainty principle?

The uncertainty principle states that there is a fundamental limit to how precisely we can know the momentum and position of a particle at the same time. The dispersion relation is closely related to this principle, as it shows that the uncertainty in the energy of a particle is inversely proportional to the uncertainty in its momentum.

## 5. Can the dispersion relation be applied to all types of particles?

Yes, the dispersion relation can be applied to all types of particles, including those with mass and those without (such as photons). However, the specific form of the dispersion relation may differ depending on the type of particle and the physical system in which it is found.

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