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lalbatros
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Let us consider a collection of non-interacting hydrogen atoms at a certain temperature T.
The energy levels of the hydrogen atom and their degeneracy are:
The partition function in statistical physics is given by:
This function is the "generating function" for all thermodynamic quantities.
The free energy is a simple function of Z.
For the spectrum of hydrogen, this partition function does not converge.
We can consider other systems:
For a particle in a box, the levels are proportional to n².
For an harmonic oscillator, the level are proportional to (n+1/2).
In both cases, the partition function converges.
The particle-in-a-box corresponds to a confined system, this might explain the convergence.
However, the harmonic oscillator is not a confined system, but the partition function does converge.
Could you help me to understand the meaning of all that?
Thanks
The energy levels of the hydrogen atom and their degeneracy are:
En = -R/n²
gn = n²
gn = n²
The partition function in statistical physics is given by:
Z = Sum(gn Exp(-En/kT), n=1 to Inf)
This function is the "generating function" for all thermodynamic quantities.
The free energy is a simple function of Z.
For the spectrum of hydrogen, this partition function does not converge.
We can consider other systems:
For a particle in a box, the levels are proportional to n².
For an harmonic oscillator, the level are proportional to (n+1/2).
In both cases, the partition function converges.
The particle-in-a-box corresponds to a confined system, this might explain the convergence.
However, the harmonic oscillator is not a confined system, but the partition function does converge.
Could you help me to understand the meaning of all that?
Thanks
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