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

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

URL: http://arxiv.org/abs/2603.02407v1
Date: Mon, 02 Mar 2026 21:33:25 GMT
Title: Collapse and transition of a superposition of states under a delta-function pulse in a two-level system
Authors: Ariel Edery,
Abstract summary: We study the transition in going from a textitlinear superposition of eigenstates to an eigenstate under a delta-function pulse.<n>With a delta-function pulse, the transitions are abrupt/instantaneous and we show that they do not depend on the energy gap.<n>We show that the system ends up in a definite eigenstate i.e. a probability of unity.
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
Abstract: 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 \textit{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 $E_1$ and $E_2$ and solve the coupled set of first order equations to obtain exact analytical expressions for the coefficients $c_1(t>0)$ and $c_2(t>0)$ 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 $α_1$ and $α_2$ of the initial superposition state which are free parameters constrained only by $|α_1|^2+ |α_2|^2=1$. This opens up new possibilities and in particular, allows for a ``collapse" scenario. We obtain a general analytical expression for the transition probability $P_{α_1,α_2 \to 2}$ 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 $E_2-E_1$ 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.

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