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Is matter still vibrating at absolute zero temperature? If it isn't, wouldn't we then be able to know both its position and momentum?

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A system at absolute zero will still have nonzero uncertainty in position and momentum.

As an example, consider the simplest vibrating system, the one dimensional harmonic oscillator with (angular) frequency $\omega$ and mass $m$. In it's ground state (i.e. the lowest possible energy for the system) the position probability amplitude is a Gaussian ('bell curve'): $$ \psi(x)\propto e^{\frac{-m \omega x^2}{2 \hbar}}$$. The finite extent of that curve reflects the position uncertainty: $\Delta x = \sqrt{\frac{\hbar}{2 m \omega}}$. One can also show that in this state the momentum uncertainty is $\Delta p= \sqrt{\frac{ m \omega \hbar}{2}}$, and it is a minimal uncertainty state ($\Delta p \Delta x = \hbar/2$).

This example is easily generalized to 3D and is actually quite relevant since vibrating systems can often be approximated quite well by a simple harmonic oscillator, especially at low energies.

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  • $\begingroup$ Are you able to say why they are still vibrating? $\endgroup$ – Shookster May 23 '14 at 3:30
  • $\begingroup$ Not sure how to answer that. Are they "vibrating" if the system is in the lowest stationary state? I would say not, but it is perhaps a matter of terminology. The wavefunction has finite extent so that there is uncertainty in position/momentum, but it is not 'moving' in any observable sense. $\endgroup$ – DrEntropy May 23 '14 at 20:48
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There will still be vibrations due to zero-point energy. So it's still impossible to know both its position and momentum.

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  • $\begingroup$ Is there any state below this zero point where the vibrations stop? $\endgroup$ – Shookster May 20 '14 at 2:57
  • $\begingroup$ No. That's why it's called zero-point. $\endgroup$ – velut luna May 20 '14 at 3:04
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You are thinking about this backwards.

Because of the uncertainty principle, zero-point energy exists.

Because of the existence of zero-point energy, it is impossible to cool any object all the way to absolute zero.

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protected by Qmechanic Jan 19 '17 at 21:54

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