# I Temperature in quantum systems

1. Apr 26, 2016

### 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!

2. Apr 26, 2016

### naima

3. Apr 26, 2016

### A. Neumaier

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.

4. Apr 26, 2016

### lfqm

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?

5. Apr 26, 2016

### naima

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.

6. Apr 26, 2016

### lfqm

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.

7. Apr 26, 2016

### naima

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 gallery of wigner functions ? 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.

8. Apr 27, 2016