# Fermi Velocity absolute zero

## Main Question or Discussion Point

Is it weird that at absolute zero in a metal, electrons at the fermi level still move around at the fermi velocity.

Is the notion that everything stops at absolute zero incorrect?

Thank you

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Kinetic energy of free electron: E=p^2/2m in classical mechanics, and E=h^2/2m in quantum mechanics.
Under the classical theory at T=0 V=0 and accordingly E=0. But in quantum mechanics at T=0 electron in a crystal has « Fermi's energy »: E=h^2(3pi^2*n)^(2/3)/2m. As you can see it does not depend on temperature.
Is the notion that everything stops at absolute zero incorrect?
Yes.
I wish success.

absolute zero and ground state

Yes. The notion that everything comes to a stop is a classical notion and quantum effects will "violate" it. Here, you see the Pauli principle in action. Even without it (i.e. for a system of bosons, or for one isolated particle), you have quantum zero-point energy, ensuring that if you measure the momentum of a particle, there is a probability that it is nonzero even at zero temperature.

The only case where the particles are strictly motionless is for a system of bosons that do not interact, or a single particle, in an infinite geometry without any potentials.

Thus: At T=0, it is not true that particles are at rest. However, it is true that the system (as a whole) is in its ground state, the state of lowest possible energy. (This is by definition, more or less.) But the ground state will typically have a nonzero probability for a particle being in motion!

Re: absolute zero and ground state

Kinetic energy of free electron: E=p^2/2m in classical mechanics, and E=h^2/2m in quantum mechanics.
Under the classical theory at T=0 V=0 and accordingly E=0. But in quantum mechanics at T=0 electron in a crystal has « Fermi's energy »: E=h^2(3pi^2*n)^(2/3)/2m. As you can see it does not depend on temperature.
I have to disagree with all the "content" of that paragraph. 1) The analogy to the energy of the free particle is completely irrelevant at this point. 2) You give a zeroth-order formula for T=0 and impose that this formula is temperature independent. That's trivial and has no significance. To the contrary, the Fermi surface gets smeered out for T>0 because there is a probability distribution in energy.

Thus: At T=0, it is not true that particles are at rest. However, it is true that the system (as a whole) is in its ground state, the state of lowest possible energy. (This is by definition, more or less.) But the ground state will typically have a nonzero probability for a particle being in motion!
I completely agree with that.