arXiv:2406.03595

Overlap integral of stationary scattering states

Kenzo Ishikawa, Y. Nishio

Published Apr 21, 2024• 0 citations

Abstract

The overlap integrals of scattering states in potentials of finite widths are expressed with their asymptotic behaviors and those of energies $E_1$ and $E_2$ consist of diagonal terms that are proportional to $δ(E_1-E_2)$ and nondiagonal terms. Owing to the composition of nondiagonal terms, superpositions of stationary states have time-dependent norms and finite probability currents. These do not represent isolate states. In various exceptional potentials and in free theory, nondiagonal terms do not exist, and the superpositions of states with different energies represent isolate particles that exactly describe scattering processes.

DOI: 10.48550/arxiv.2406.03595

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arXiv:2406.03595

Overlap integral of stationary scattering states

Kenzo Ishikawa, Y. Nishio

Published Apr 21, 2024• 0 citations

arXiv HTML PDF

Abstract

The overlap integrals of scattering states in potentials of finite widths are expressed with their asymptotic behaviors and those of energies

E_1

and

E_2

consist of diagonal terms that are proportional to

δ(E_1-E_2)

and nondiagonal terms. Owing to the composition of nondiagonal terms, superpositions of stationary states have time-dependent norms and finite probability currents. These do not represent isolate states. In various exceptional potentials and in free theory, nondiagonal terms do not exist, and the superpositions of states with different energies represent isolate particles that exactly describe scattering processes.

DOI: 10.48550/arxiv.2406.03595

Related papers

Potential scatterings in $L^2$ space: (1) non-orthogonality of stationary statesPotential scatterings in $L^2$ space: (1) non-orthogonality of stationary states

Kenzo Ishikawa

Orthogonality of eigenstates of different energies and its implications in potential scattering are unlabeled. Scalar products of scattering states of different energies are found to have finite non-orthogonal terms in potentials of finite widths. Their superpositions have time-dependent norms, and are not suitable for isolate states. In these systems, a perturbative method and a variational method are viable methods for finding a rigorous transition probability that describes phenomena completely. In various exceptional potentials, an orthogonality is satisfied.Orthogonality of eigenstates of different energies and its implications in potential scattering are unlabeled. Scalar products of scattering states of different energies are found to have finite non-orthogonal terms in potentials of finite widths. Their superpositions have time-dependent norms, and are not suitable for isolate states. In these systems, a perturbative method and a variational method are viable methods for finding a rigorous transition probability that describes phenomena completely. In various exceptional potentials, an orthogonality is satisfied.

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The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain (Asymptotic Condition, in- and out- states, scattering operator $\hat{S}$ ). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as $| ψ_{E(\pm 1)}^\pm \rangle$ ), to discuss fundamental properties of $| ψ_{E(\pm 1)}^\pm \rangle$ , and subsequently to construct $\hat{S}$ and $\hat{T}$ operators in terms of $| ψ_{E(\pm 1)}^\pm \rangle$ . Physical contents of the $\hat{S}$ and $\hat{T}$ operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain (Asymptotic Condition, in- and out- states, scattering operator $\hat{S}$). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as $| ψ_{E(\pm 1)}^\pm \rangle$), to discuss fundamental properties of $| ψ_{E(\pm 1)}^\pm \rangle$, and subsequently to construct $\hat{S}$ and $\hat{T}$ operators in terms of $| ψ_{E(\pm 1)}^\pm \rangle$. Physical contents of the $\hat{S}$ and $\hat{T}$ operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.

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Orthogonality Catastrophe for a System of Interacting Electrons. II

K. Yamada, Kei Yosida

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Scattering

Abstract Besides the bounded, periodic classical motion of particles in potentials as shown in Fig. 5.2 (for energy E1 ), there is also the possibility of unbound states which are not localized in space and not repetitive in time; these correspond to particles incident on the potential and which subsequently “bounce “ off (energy E2 in Fig. 5.2) or temporarily change their speeds as they go over the potential (energy E3 ). Classical mechanics, which solves for the exact trajectories in either case, makes little distinction between the two types of motion aside from some tech- nical details,Abstract Besides the bounded, periodic classical motion of particles in potentials as shown in Fig. 5.2 (for energy E1 ), there is also the possibility of unbound states which are not localized in space and not repetitive in time; these correspond to particles incident on the potential and which subsequently “bounce “ off (energy E2 in Fig. 5.2) or temporarily change their speeds as they go over the potential (energy E3 ). Classical mechanics, which solves for the exact trajectories in either case, makes little distinction between the two types of motion aside from some tech- nical details,

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Transition of a superposition of states under a delta-function pulse in a two-level system

Ariel Edery

Under a time-dependent perturbation, it is common to calculate the probability of a transition from one eigenstate to another eigenstate of a quantum system. Here we study the transition from a \textit{linear superposition of eigenstates} to an eigenstate under a delta-function pulse. We consider a two-level system with energy levels $E_1$ and $E_2$ and obtain exact analytical expressions for the coefficients $c_1$ and $c_2$ of the final state. The expressions are general since the coefficients $α_1$ and $α_2$ of the initial superposition state are free parameters constrained only by $|α_1|^2+ |α_2|^2=1$ . This opens up new possibilities and in particular allows for an abrupt transition to a definite eigenstate with unit probability. We obtain a general analytical expression for the probability $P_{α_1,α_2 \to 2}$ of an initial superposition state to transition 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 values of the interaction strength $β$ , the initial superposition transitions abruptly to a definite eigenstate with unit probability. We discuss the similarities and differences such a transition has with the collapse of the wavefunction familiar in the context of a measurement.Under a time-dependent perturbation, it is common to calculate the probability of a transition from one eigenstate to another eigenstate of a quantum system. Here we study the transition from a \textit{linear superposition of eigenstates} to an eigenstate under a delta-function pulse. We consider a two-level system with energy levels $E_1$ and $E_2$ and obtain exact analytical expressions for the coefficients $c_1$ and $c_2$ of the final state. The expressions are general since the coefficients $α_1$ and $α_2$ of the initial superposition state are free parameters constrained only by $|α_1|^2+ |α_2|^2=1$. This opens up new possibilities and in particular allows for an abrupt transition to a definite eigenstate with unit probability. We obtain a general analytical expression for the probability $P_{α_1,α_2 \to 2}$ of an initial superposition state to transition 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 values of the interaction strength $β$, the initial superposition transitions abruptly to a definite eigenstate with unit probability. We discuss the similarities and differences such a transition has with the collapse of the wavefunction familiar in the context of a measurement.

quant-pharXiv:2603.02407Physics Letters AMar 4, 20260 citations

The overlap integrals of scattering states in potentials of finite widths are expressed with their asymptotic behaviors and those of energies

E_1

and

E_2

consist of diagonal terms that are proportional to

δ(E_1-E_2)

Potential scatterings in

L^2

space: (1) non-orthogonality of stationary states

\hat{S}

). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as

| ψ_{E(\pm 1)}^\pm \rangle

), to discuss fundamental properties of

| ψ_{E(\pm 1)}^\pm \rangle

, and subsequently to construct

\hat{S}

and

\hat{T}

operators in terms of

| ψ_{E(\pm 1)}^\pm \rangle

. Physical contents of the

\hat{S}

and

\hat{T}

operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.

Asymptotes of non-stationary solutions of the Schrödinger equation for a particle interacting with a one-dimensional

δ

-potential

t\to\mp\infty

; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schrödinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.

E_1

and

E_2

and obtain exact analytical expressions for the coefficients

c_1

and

c_2

of the final state. The expressions are general since the coefficients

α_1

and

α_2

of the initial superposition state are free parameters constrained only by

|α_1|^2+ |α_2|^2=1

. This opens up new possibilities and in particular allows for an abrupt transition to a definite eigenstate with unit probability. We obtain a general analytical expression for the probability

P_{α_1,α_2 \to 2}

of an initial superposition state to transition 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 values of the interaction strength

β

, the initial superposition transitions abruptly to a definite eigenstate with unit probability. We discuss the similarities and differences such a transition has with the collapse of the wavefunction familiar in the context of a measurement.