Collapse and transition of a superposition of states under a delta-function pulse in a two-level system

Under a time-dependent perturbation it is common to calculate the transition probability in going from from one eigenstate to another eigenstate of a quantum system. In this work we study the transition in going from a linear superposition of eigenstates to an eigenstate under a delta-function pulse which acts at $t=0$. We consider a two-level system with energy levels E₁ and E₂ and solve the coupled set of first order equations to obtain exact analytical expressions for the coefficients c₁(t>0) and c₂(t>0) of the final state. The expressions for the final coefficients are general in the sense that they are functions of the interaction strength β and the coefficients α₁ and α₂ of the initial superposition state which are free parameters constrained only by |α₁|²+ |α₂|²=1. This opens up new possibilities and in particular, allows for a "collapse" scenario. We obtain a general analytical expression for the transition probability P_{α1,α2 → 2} in going from an initial superposition state to the second eigenstate. Armed with this general expression we study some interesting special cases. With a delta-function pulse, the transitions are abrupt/instantaneous and we show that they do not depend on the energy gap E₂-E₁ and hence on the relative phase between the two eigenstates. For specific multiple values of the interaction strength β, we show that the system ends up in a definite eigenstate i.e. probability of unity. Such a transition can be viewed as a "collapse" since a superposition of states transitions abruptly to a definite eigenstate. The collapse of the wavefunction is familiar in the context of a measurement. Here it occurs via a delta-function pulse in Schrödinger's equation. We discuss how this differs from a collapse due to a measurement.