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I Temperature in quantum systems

  1. Apr 26, 2016 #1
    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. jcsd
  3. Apr 26, 2016 #2

    naima

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  4. Apr 26, 2016 #3

    A. Neumaier

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    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.
     
  5. Apr 26, 2016 #4
    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?
     
  6. Apr 26, 2016 #5

    naima

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    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.
     
  7. Apr 26, 2016 #6
    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.
     
  8. Apr 26, 2016 #7

    naima

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    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.
     
  9. Apr 27, 2016 #8

    naima

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    In this old thread
    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.
     
  10. Apr 27, 2016 #9

    A. Neumaier

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    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.
     
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