Fermi's Golden Rule

Question

Consider a system with countable quantum states. One can define $J_{ij}$ as the rate of transition of probability from i-th to j-th quantum state. In H-theorem, if one assumes both $$ H:=\sum_{i} p_{i}\log p_{i}$$ $$J_{ij}=J_{ji}$$ 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:$$\mathrm{i}\hbar\frac{\mathrm{d}\psi}{\mathrm{d}t}=H\psi$$ 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 $\psi_{i}$.) $$J_{ij}\propto |\langle{\psi_{j},H\psi_{i}}\rangle|^2=\langle{\psi_{j},H\psi_{i}} \rangle\overline{\langle{\psi_{j},H\psi_{i}}\rangle}$$ and H is Hermitian, so: $$J_{ij}\propto |\langle{\psi_{j},H\psi_{i}}\rangle|^2=\overline{\langle{H\psi_{j},\psi_{i}} \rangle}\langle{H\psi_{j},\psi_{i}}\rangle=\langle{\psi_{i},H\psi_{j}} \rangle\overline{\langle{\psi_{i},H\psi_{j}}\rangle}$$ Hence: $$J_{ij}=J_{ji}$$

As a result, we can prove that entropy for an isolated system always increases 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?

Answer 1

Something important is missing from your presentation of the Fermi rule. The Schroedinger eq. you mention, $i\hbar(d\psi/dt) = H\psi$, is never going to produce any transitions between eigenstates of $H$ itself: by definition eigenstates are stationary states.

What you probably refer to is something like $$ i\hbar\frac{\partial \psi_I}{\partial t} = V_I(t) \psi_I $$ which is the interaction picture form of the Schroedinger eq. in the presence of a perturbation $V$, $i\hbar(d\psi/dt) = (H + V)\psi$. Here $\psi_I(t) = e^{(i/\hbar)Ht}\psi(t)$ and $V_I(t) = e^{(i/\hbar) H t} V e^{-(i/\hbar) H t}$. If $\psi_{i(j)}$ are eigenstates of $H$, then Fermi's golden rule indeed gives the transition rate between them to 2nd order in the (small) perturbation $V$: $$ J_{i\rightarrow j} \sim |\langle \psi_j|V|\psi_i\rangle|^2 $$ In other words, Fermi's rule concerns an open system undergoing a weak interaction with its environment, usually represented by an electromagnetic field, or more generally, by a an external thermodynamic "bath".

This being said, Fermi's rule is known to be equivalent to the Markov approximation for open systems, see R.Alicki, "The Markov master equations and the Fermi golden rule", Int.J.Theor.Phys.Vol.16(5), 351-355(1977). A very important consequence of the Markov approximation is that the dynamics is no longer time reversible: while under the original Hamiltonian dynamics entropy is conserved, under the Markov approximation it is not. In fact, as you point out, it is possible to justify the H-theorem, and under some additional conditions it can be shown that the dynamics is governed by a dynamical semigroup with a Lindblad-type generator (see for instance Sec.IIC on the "Secular Approximation" here).

Now to the actual question: Fermi's rule is definitely not a principle. The Markov approximation, and therefore Fermi's rule, holds provided the relaxation time $\tau$ it describes for the system is much longer than the bath relaxation time, $\tau >> \tau_{bath}$. When the system-bath interaction is too strong and/or the time scale for the system's relaxation becomes comparable to that of the bath, both the Markov approximation and Fermi's rule cease to apply. What this means is that the dynamics is no-longer memoryless, but depends on the past history of the system. The transition from Markovian to non-Markovian dynamics can be seen even in such a simple system as a qubit in a dissipative environment, which makes it important for entanglement and decoherence problems. For instance, a qubit undergoing revival of coherence driven by a dissipative bath no longer follows a Markovian dynamics. See for instance this recent review on "Non-Markovian dynamics in open quantum systems".