Undergrad Temperature in quantum systems

[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!

 

[naima]

Read thermal coherent states in the wiki. I found it enlightning.

 

[A. Neumaier]

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.

 

[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?

 

[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.

 

[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.

 

[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 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]

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

In this old thread

Let's first focus on the idea of a single particle in a quantum mechanical system which is set at some temperature T. When a system is at some temperature T it means that its energy is not fixed. The particle can sit in each energy eigenstate, and the probability that it does so is given by $e^{-E/(kT)}/Z$

This would mean that a Fock state with a well defined energy is not at a precise temperature. This is the case with the harmonic oscillator hamiltonian.

 

[A. Neumaier]

a Fock state with a well defined energy is not at a precise temperature.

Conventionally, Fock states are considered as (excited) zero temperature states. It is impossible to assign it a meaningful nonzero temperature, not even an imprecise one.