Given the value of my observable (Energy) and with a known hamiltonian, I would like to compute the value of my thermal state.
That is, with a given observable defined as: $$\langle\hat{O}\rangle=tr(\hat{O}\rho),$$
where the density matrix is
$$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$$
and
$$p_{i}=\frac{e^{-\beta{E_{i}}}}{Z}.$$
Given my observable value and the Hamiltonian, I want to know the value of $\beta.$
Is there any direct way to do it? How can I compute it in MATLAB?
Thank you very much in advance for your help and support.