Partition function and Quantum mechanics

[eljose]

Let be the Hamiltonian Energy equation:

$$H\Psi= E_{n} \Psi$$

then let be the partition function:

$$Z=\sum_{n} g(n)e^{-\beta E_{n}}$$

where the "Beta" parameter is 1/KT k= Boltzmann constant..the question is..let,s suppose we know the "shape" of the function Z...could we then "estimate" the Hamiltonian that yields to these energies?..thanks.

 

[lalbatros]

eljose,

Finding out the energy spectrum and the g factors from Z is clearly possible in principle.

This is very similar to the spectrum analysis of a signal.
Numerical methods will be simple curve fitting, this would be similar to finding out several decay factors in a signal, like in nuclear metrology or in other sciences (like biology).

For analytical considerations, it could be useful to consider first how you would solve the analog problem for Z(i beta) instead of Z(beta): this would bring you back into the Fourier analysis domain. Maybe you don't need this flashback if you know the Laplace transforms better then I do. Maybe a Laplace inversion is all that you need.

Finally from the energy spectrum and g factor, you can write H as a diagonal matrix and you are done -at first sight-. What you will still be missing is a physical representation of the Hilbert space supporting this Hamiltonian. I mean that this Hamiltonian - a diagonal matrix- will not offer you much insignt into the physics. Assume for example you are dealing here with the Hydrogen atom. Having found out the hamiltonian will not tell you how to modify it to include an external electric field. However, repeating this "Z procedure" for several electric fields will show you how the spectrum and g's are modifies under the Stark effect.

Finally, I wonder if this method has not already be used somehow. Think for example to the specific heat of solids. The analysis of some thermodynamic data might have given already some information on the interactions (hamiltonian). Do you know some examples?

Michel

 

[eljose]

Of course we have that...$$Z(\beta)=Tr[e^{-\beta H}]$$ but from this equation we couldn,t get the Hamiltonian...

If we consider the continuum approach...

$$Z(\beta)=\int_{-\infty}^{\infty}dpdqe^{-\beta[p^{2}+V(x)}]$$

integrating over the "p" variable and knowing Z you get a Non-linear integral equation...the problem is that i don,t know how to solve it to get V(x) only a Fourier-based method to get the inverse so V(f(x))=x