Collapse and transition of a superposition of states under a delta-function pulse in a two-level system
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908861402710016 |
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| author | Edery, Ariel |
| author_facet | Edery, Ariel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02407 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Collapse and transition of a superposition of states under a delta-function pulse in a two-level system Edery, Ariel Quantum Physics High Energy Physics - Theory 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. |
| title | Collapse and transition of a superposition of states under a delta-function pulse in a two-level system |
| topic | Quantum Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2603.02407 |