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I have long wondered why nobody seems to think that this form of the uncertainty principle proves it impossible to reach absolute zero.

I noticed that there is a proof suggesting it requires infinite time to reach absolute zero in A general derivation and quantification of the third law of thermodynamics by Masanes and Oppenheim:

"Here, we provide a derivation of the principle that applies to arbitrary cooling processes, even those exploiting the laws of quantum mechanics or involving an infinite-dimensional reservoir. We quantify the resources needed to cool a system to any temperature, and translate these resources into the minimal time or number of steps, by considering the notion of a thermal machine that obeys similar restrictions to universal computers. We generally find that the obtainable temperature can scale as an inverse power of the cooling time."

But they do not mention a connection to the uncertainty principle. Can anyone help?

Conifold
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G.Malllett
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1 Answers1

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The Energy-Time uncertainty principle is very different to the position-momentum uncertainty principle. $\Delta E$ is the uncertainty in energy, as you might expect. $\Delta t$ is the minimum time at which we would expect any measurable property of the system to have changed 'significantly'. Specifically $$ \Delta t =\frac{\Delta B} {|\frac{d<B>} {dt} |} $$ so the amount of time for some measurable to change by its expectation value.

Thus the conclusion we draw is the opposite of what you propose in your question, we are certain of the value of the energy so $\Delta E \approx 0$ so $\Delta t$ is massive and it will take a very long time for the quantum system to evolve out of its current state without outside input.

The reason for absolute zero being difficult, if not impossible, to reach is that its entropy is zero, and the macroscopic statistical 'laws' of nature require the entropy of an entire system to increase. To cool something down something else must be heated up more to have an overall increase in entropy. As we get close to absolute zero, the required increase in temperature of the other object gets very, very large, and it becomes unfeasible to continue.

Eddy
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