In time-dependent perturbation theory the main goal is to determine the time-evolution of a perturbed quantum system, with particular emphasis on calculating transition probabilities and modeling the irreversible decay of probability from a small quantum system coupled to a very large quantum system. The interaction picture will be used to discuss time-dependent perturbation.
Time Dependent Perturbation Theory
Come back to Schrödinger picture. Suppose at t=0 we works out the eigenstates ∣n⟩ of H0 . We are interested in two of them, ∣f⟩ and ∣i⟩ (initial state and final state in Schrödinger picture). After time t , the states in the Schrödinger picture become
The questions is, what the transition probability Pfi(t) is for a system initially in the state ∣i⟩ at time t=t0 (could be zero) evolving into the state ∣f,t⟩ at time t ? Using the evolution operator, we can derive
Notice that the sinc-squared function sequence πN(Nx)2sin2(Nx) approaches the behavior of delta function when N→∞. We naturally introduce a distribution δT(E) as follows
which has the maximum value 2πℏT at E=0 and its width of order is T2πℏ. It becomes delta function when T→∞. Therefore, the transition probability can be written as
For continuum, the states are labeled by the energy E and some other parameters α, and a significant concept in continuum is the density of state (DOS) ρ(α,E).
We shall calculate the transition probability of a system after time T under a sinusoidal perturbation with frequency ω . From previous deductions, we can foresee that the system will be most likely to transit to those states with eigenenergy Ef=Ei+ℏω. However, the eigenvalue Ef might be degenerated, containing a variety of states ∣f,Ef⟩ satisfying the condition, while their matrix elements ⟨f,Ef∣W∣i⟩ might be different even though they share the same energy. This makes the problem intractable.
Anyway, we start from the very beginning. The probability of transition is
If Ef has finite-fold degeneracy, the integration over f will be replaced by summation
where Ef=Ei+ℏω . This is known as Fermi’s Golden Rule.
First, we use the first-order time dependent theory, which requires that T is small enough without making transition probability P>1. The characteristic time is given by
Also, we approximate the sinc-squared function sequence as delta function, which requires that T is large. But how to evaluate the order of magnitude of T ? We hope T is large enough to make T2πℏ, the width of δT(E), relatively small compared to the energy range of the system ΔE , so that δT(E) can pick out a narrow range around Ef . The latter is given by the width of
ΔE=width of K(E)=width of ∫df∣⟨f,E∣W∣i⟩∣2ρ(f,E)
where K(E) is a function that evaluates the interaction between initial states ∣i⟩ and states with different energies E.
Thus the second condition is
Therefore we have ΔE≫Γ.
In order to study the evolution of the system on a timescale that could be long compared to Γ−1 one must use a non-perturbative method. The means of solving this type of problem was introduced in 1930 by Weisskopf and Wigner.
The Schrodinger equation version of time-dependent perturbation theory actually does the following things: Assume ∣ψ(t)⟩=∑bne−iEnt/ℏ∣φn⟩, and then obtain
And do Taylor expansion, compare terms with the same order:
The second equals sign comes from W only have non-diagonal matrix element. Notice that, all calculations are strictly accurate by now. If we separately consider the evolution of initial state ∣i⟩ and the other states ∣k⟩, the ODE with be
In the second equation, we do an approximation that only the interaction between the initial state and other states are considered. The interactions within other state themselves are neglected. We can then substitute the second equation into the first one:
Assume that the derivative b˙i(t) has only a very short memory of the previous values of b(t′) between 0 and t . Actually, it depends only on the values of at times immediately before. So we can substitute b(t′) with b(t). And since t is not the integration variable, it can be extracted out.
For the first term, with Γ=2ℏπK(Ei+ℏω), that is actually 2Γ! For the second term, the factor Ef−(Ei+ℏω)1 actually picks out the effect of K(E) around Ei+ℏω. Notice that K(E) has a unit of energy, we define the integral as E. The evolution of bi(t) is thus described with
which implies the exponential decay
Distribution of Final State
Substitute bi(t)=e−Γt/2e−iEt/ℏ into iℏdtdbk(t)≈eiωkitWki(t)bi(t), we find: