# Temperature in quantum systems

• I
• lfqm
In summary, the conversation focuses on studying thermodynamics of quantum systems, specifically the hamiltonians of quantum mechanics at different temperatures. It is discussed that at zero temperature, the ground state is a gaussian with minimal width, and at finite temperature, it becomes a thermal coherent state. The concept of temperature for a Fock state is also debated, with the conclusion that it cannot be assigned a meaningful nonzero temperature.

#### lfqm

Hi!

These days I've been studying thermodynamics of quantum systems, and in so a very basic doubt come to me... I hope you guys can help me:

When we study the usual hamiltonians of quantum mechanics (H-atom, harmonic oscillator, etc.)... Are these hamiltonians modeling the system at temperature 0? How can the temperature be adjusted in the hamiltonian?

More concretely: How do I study a quantum harmonic oscillator at temperature 0 and how do I do it at finite temperature?

Thanks!

lfqm said:
How do I study a quantum harmonic oscillator at temperature 0 and how do I do it at finite temperature?
At zero temperature by looking at wave functions (ground states and excited states), at finite temperature by looking at density operators - in the simplest case canonical ensembles.

So, the hamiltonian isn't modified?
At 0 temperature I study the usual spectrum of the hamiltonian and at finite temperature I use the density operador given in statistical mechanics?

The hamiltonian is the same. At zero temperature the ground state is a gaussian with a minimal width (a coherent state). if you enlarge it, you get the ground state at a finite temperature (it is then a thermal coherent state). If you translate it in the phase space the temperature does not change but you are no more in the vacuum.
Inside a black body you have a thermal coherent state It is well explained in the wiki link.

Ok, at 0 tenperature the ground state is a coherent state with Alfa=0... But, the first excited state at 0 temperature is the fock state |1>? i.e. the usual spectrum.

As the single-particle Fock state ##|1 \rangle## has a greater energy than the ground state it is an excitation but it is not a thermal coherent state. Have you seen this http://www.iqst.ca/quantech/wiggalery.php ? I am not sure that there is a notion of temperature for this state. You are talking about the first excited state. if you translate the vacuum by a small complex ##\alpha## you get a coherent state with an energy which can be less than the energy of ##|1 \rangle##
I recently discovered this notion of thermal quantum state please correct my eventual errors.

naima said:
I am not sure that there is a notion of temperature for this Fock state.