I asked a question regarding "temperature at quantum level" here:
[What is temperature on a quantum level?](https://physics.stackexchange.com/questions/491179/what-is-temperature-on-a-quantum-level)

I didn't get the answer i got and i have yet another question regarding to the elusive temperature.

I know that temperature is a statistical mechanics term, and does not apply to quantum mechanics.

We learned, since high school, but even in modern Classical Mechanics courses, that temperature (whatever that is), is related to the kinetic energy of the atoms of a system.

If that was the case, then why isn't a temperature term added to the Hamiltonian of the system?

Why when we calculate the Hamiltionian, we place a Kinetic Energy term and a Potential Energy term.

Why is the temperature of the system that the atom is placed in, never taken into account, and placed as a Kinetic Energy term in the Hamiltonian?