Quantum description of steady-state processes

The time-dependent Schrodinger equation can be evaluated to yield stationary solutions of the form 

In classical physics we are familiar with another kind of stationary states, so-called steady states, for which observables are still constant in time however fluxes do exist. A system can asymptotically reach such a state when the boundary conditions are not compatible with equilibrium, for example, when it is put in contact with two heat reservoirs at different temperatures or matter reservoirs with different chemical potentials. Classical kinetic theory and nonequilibrium statistical mechanics deal with the relationships between given boundary conditions and the resulting steady-state fluxes. The time-independent formulation of scattering theory is in fact a quantum theory of a similar nature (see Section 2.10). 

In addition to studying actual steady-state phenomena, it is sometime useful to use them as routes for evaluating rates. Consider, for example, the first-order reaction A — P and suppose we have a theory that relates A(t) to A(t = Q). A rate coefficient can then be defined by k(t) = —ln [ (Z = 0)] -4(Z), though its usefulness is usually limited to situations where k is time-independent, that is, when A obeys first-order kinetics, A t), at least for long times. In the latter 

What is the quantum mechanical analog of this approach Consider the simple example that describes the decay of a single level coupled to a continuum. Fig. 9.1 and Eq. (9.2). The time-dependent wavefunction for this model is (Z) = Ci (t) 11)+ Ci(t ) l, where the time-dependent coefficients satisfy (cf. Eqs (9.6) and (9.7)) 

The result (9.24) is obtained by solving this as an initial value problem, given that Ci(Z = 0) = 1. Alternatively, suppose that the population in state 1) remains always constant so that Ci(t) = ci exp(—In this case the first equation in (9.67) is replaced by Eq. (9.68a) below, where we have also supplemented the second equation by an infinitesimal absorbing term, so that 

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