# Microstates/ Eigenstates of Hamiltonian and multiplicity

I have 2 specific questions regarding micro-states, entropy and the relation with the hamiltonian.

1. If we observe a closed system, then the microstates of it are eigenstates of the hamiltonian. Is there a way to prove this? We took it as a fact in our class, but it wasn't proven.

2. When the closed system goes in equilibrium, the macrostate in which the system is found, is the one with the highest multiplicity, highest nr. of microstates. At the same time, we know that every system tries to reach stability by having the lowest possible energy. This implies that the macrostate of equilibrium, is the state with the lowest energy and the highest multiplicity. Is there a correlation between the two? Is there a mathematical/physical way to prove this, if it is true?

• The system tries to minimize the free energy. This means there is a competition between minimizing the internal energy and maximizing the multiplicity. Nov 17, 2021 at 22:13
• Why a competition? Does multiplicity maximization results in increase of internal energy? Nov 17, 2021 at 22:37
• Consider $F = U - TS$. At zero temperature you minimize $U$. At infinite temperature you maximize $S$. In between it is neither of these two extremes. Nov 17, 2021 at 23:51
• And regarding my initial question. Can I find such a proof? Dec 11, 2021 at 23:17

1. Yes it is provable. For a good derivation, I recommend reading up on it in the textbook, A. V. Rao, “Dynamics of Particles and Rigid Bodies: A Systematic Approach”, Cambridge Press.

2. No.

Hope this helps! :)