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I remember my Physical Chemistry Professor saying that very tiny negative Kelvin temperatures have been achieved on the quantum level. Is this true?

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Welcome to @zero. This is an interesting question, but it may be more appropriate at – Ben Norris Nov 9 '12 at 11:35
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Yes. But it is a bit of a joke of a sort. In any system with energy levels, the lower energy levels never have a lower population than the higher ones. In fact there is an equilibrium arrangement of the populations of such levels characterized by the temperature. The higher the temperature, the more evenly populated the levels are. There is even a formula for calculating the populations in this case.

The energy levels of some systems depends on a magnetic field. You can let the populations of the energy levels come to equilibrium at some temperature $T$ and then quickly invert the magnetic field. What was a low energy level now becomes a high one and vice versa. You have an upside down distribution of populations.

Soon enough that upside down population decays to a right side up population.

But while the distribution is "upside down" one finds that it is given by the same formula used for an ordinary distribution, but with a negative temperature.

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Here is my rather easy explanation for negative energies, using the mentioned equation: – AstronAUT Sep 5 '15 at 18:22

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