- #1
Kiarash
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Consider a system with countable quantum states. One can define Jij as the rate of transition of probability from i-th to j-th quantum state. In H-theorem, if one assumes both $$ H:=\Sigma_{i} p_{i}log(p_{i})$$ $$J_{ij}=J_{ji}$$ then they can prove the H always decrease. The latter is Fermi's Golden Rule states that the transition rate's matrix is symmetric.
I have seen in Federick Reif's book Fundamentals of Statistical and Thermal Physics, he has proven Fermi's rule. Briefly, consider a quantum system which obeys Schrödinger's equation:$$i\hbar (d/dt)(\psi)=H\psi$$
where H is Hermitian. Then one can use these relations to _prove_ Fermi's Golden Rule in this specific case: (I show i-th eigenvector with Ψi.)
$$J_{ij}\alpha |\langle{\psi_{j},H\psi_{i}}\rangle|^2=\langle{\psi_{j},H\psi_{i}} \rangle\overline{\langle{\psi_{j},H\psi_{i}}\rangle}$$ and H is Hermitian, so: $$J_{ij}\alpha |\langle{\psi_{j},H\psi_{i}}\rangle|^2=\overline{\langle{H\psi_{j},\psi_{i}} \rangle}\langle{H\psi_{j},\psi_{i}}\rangle=\langle{\psi_{i},H\psi_{j}} \rangle\overline{\langle{\psi_{i},H\psi_{j}}\rangle}$$ Hence:
$$J_{ij}=J_{ji}$$
As a result, we can proof entropy for an isolated system always increase at least for some special cases with these assumptions:
I. If our quantum states are countable.
II. If our system can be described with a Hamiltonian that is Hermitian.
I have a question: do you have an example of a system does not obey these two assumptions? If so, is Fermi's Golden Rule a principle? How can we prove it using quantum mechanics? Do you know some articles about it?
I have seen in Federick Reif's book Fundamentals of Statistical and Thermal Physics, he has proven Fermi's rule. Briefly, consider a quantum system which obeys Schrödinger's equation:$$i\hbar (d/dt)(\psi)=H\psi$$
where H is Hermitian. Then one can use these relations to _prove_ Fermi's Golden Rule in this specific case: (I show i-th eigenvector with Ψi.)
$$J_{ij}\alpha |\langle{\psi_{j},H\psi_{i}}\rangle|^2=\langle{\psi_{j},H\psi_{i}} \rangle\overline{\langle{\psi_{j},H\psi_{i}}\rangle}$$ and H is Hermitian, so: $$J_{ij}\alpha |\langle{\psi_{j},H\psi_{i}}\rangle|^2=\overline{\langle{H\psi_{j},\psi_{i}} \rangle}\langle{H\psi_{j},\psi_{i}}\rangle=\langle{\psi_{i},H\psi_{j}} \rangle\overline{\langle{\psi_{i},H\psi_{j}}\rangle}$$ Hence:
$$J_{ij}=J_{ji}$$
As a result, we can proof entropy for an isolated system always increase at least for some special cases with these assumptions:
I. If our quantum states are countable.
II. If our system can be described with a Hamiltonian that is Hermitian.
I have a question: do you have an example of a system does not obey these two assumptions? If so, is Fermi's Golden Rule a principle? How can we prove it using quantum mechanics? Do you know some articles about it?