I'll save you some pain, if [math]T[/math] goes to zero, then there can be no energy present at all. It is a general fact of the work of Planck and others, that the energy of a system is roughly proportional to its thermodynamic properties. In classical mechanics, an expectation of a vacuum state is

[math]<0|H|0> = 0[/math]

This is a ground state, no energy present and no temperature can be present, system indicating a perfect vacuum. In quantum mechanics however, this is not true and the expectation value is

[math]<0|H|0> \ne 0[/math]

This is because of vacuum fluctuation motion, and motion is the approximation to a temperature. This is why a system cannot ever reach a zero temperature state, because Plancks law was shown to require a correction term to prevent temperature going to zero

[math]E = \hbar \omega \frac{1}{e^{\frac{\hbar \omega}{kT}} - 1} + \frac{1}{2}\hbar \omega[/math]

This means the fluctuation is in fact contributing a minimal temperature to the vacuum.

**Edited by Dubbelosix, 17 January 2019 - 11:08 AM.**