Evolution after switching a perturbation

Until a coupling of the system with an electromagnetic field is established, the excited states have an infinite lifetime. However, in reality the excited states have a fimte lifetime, emit photons, and as a result the energy of the system is lowered (although together with the photons the energy remains constant). Quantitative description of spontaneous photon emission has been given by Einstein. 

The most mysterious feature of the Schrodinger equation (2.13) is its linear character. The world is non-linear, because effect is never strictly proportional to cause. However, if i/ i(jc, t) and i/ 2(Jc, t) satisfy the time dependent Schrodinger equation, then their arbitrary linear combination also represents a solution.  

Let us suppose that we have a system with the Hamiltonian and its stationary 

Let us assume, that at time t = 0 the system is in the stationary state At r = 0 a drama begins one switches on the perturbation V(x,t) that in general depends on all the coordinates (x) and time (r), and after time t the perturbation is switched off. Now we ask question about the probability of 

After the perturbation is switched on, the wave function is no longer stationary and begins to evolve in time according to the time-dependent Schrodinger equation +V)ip = ih. This is a differential equation with partial derivatives with the boundary condition (x, t = 0) = The functions form 

---

If one is to apply first-order perturbation theory, two things have to be assured the perturbation V has to be small, and the time of interest has to be small (switching the perturbation in corresponds to r = 0). This is what we are going to assume from now on. At t = 0. one starts from the mth state and therefore c, = 1, while other coefficients c = 0. Let us assume that to the first approximation, the domination of the mth state continues even after switching the perturbation on. and we will be interested in detecting the most important tendencies in time evolution of c for n ft m. These assumptions (they give first-order perturbation theory ) lead to a considerable simplification of Eqs. (2.21) ... 

Its poles are determined to any order of by expansion of M. However, even in the lowest order in the inverse Laplace transformation, which restores the time kinetics of Kemni, keeps all powers to Jf (t/xj. This is why the theory expounded in the preceding section described the long-time kinetics of the process, while the conventional time-dependent perturbation theory of Dirac [121] holds only in a short time interval after interaction has been switched on. By keeping terms of higher order in i, we describe the whole time evolution to a better accuracy. 

In (11.8) Xba is the response coefficient relating a static response in (5) to a static perturbation associated with a field F which couples to the system through an additive term H = —FA in the Hamiltonian. Consider next the dynamical experiment in which the system reached equilibrium with Hq - - H and then the field suddenly switched off. How does A(S), the induced deviation of B from its original equilibrium value (B)o, relax to zero The essential point in the following derivation is that the time evolution is carried out under the Hamiltonian Hq (after the field has... 

---