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Scattering a particle on a one-dimensional $\delta$-potential barrier: asymptotic superselection rule

Nikolay Leonidovich Chuprikov

Published May 3, 2018• 0 citations
OpenAlex

Abstract

As is stated, the modern mathematical ($C^*$-algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the $\delta$-potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit $t\to \infty$, when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.

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Scattering a particle on a one-dimensional δ\deltaδ-potential barrier: asymptotic superselection rule
As is stated, the modern mathematical (C∗C^*C∗-algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the δ\deltaδ-potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit t→∞t\to \inftyt→∞, when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.
It is shown that the Stone-von Neumann theorem is inapplicable to scattering a quantum nonrelativistic particle on a one-dimensional "short-range" potential barrier, since the unboundedness of the position operator plays here a crucial role. The Shcrödinger representation associated with this process is reducible: long before and long after the scattering event the space of its asymptotes represents the direct sum of the subspaces of left and right asymptotes. There is a dichotomous-context-induced superselection rule (SSR), in which the role of a superselection operator is played by the Pauli matrix σ3σ_3σ3​ and the role of superselection (coherent) sectors is played by the above subspaces. By the SSR any superposition of states from different coherent sectors is a mixed state, and splitting the incident wave packet into the transmitted and reflected parts is nothing but a conversion of a pure state into a mixed one. The average values of any observable can be defined only for the transmission and reflection subprocesses. The former evolves within a single coherent sector of the Hilbert space, in the momentum representation; while the latter evolves within a single coherent sector of this space, in the coordinate representation.
Spontaneous symmetry breaking in the problem of scattering a particle by a one-dimensional δ\deltaδ-potential barrier
It is shown that spontaneous symmetry breaking occurs in this scattering problem. As a result, the unitary, asymptotically free quantum dynamics of this process is unstable. It transforms into dynamics associated with the violation of PPP- and TTT-symmetries and the appearance of a superselection rule. Here, the non-unitary scattering state is a mixed vector state, which is a coherent superposition of transmission and reflection states related to different Hamiltonians. Superselection operators for the transmission and reflection subprocesses are defined.
Asymptotes of non-stationary solutions of the Schrödinger equation for a particle interacting with a one-dimensional δδδ-potential
It is shown that non-stationary solutions of the Schrödinger equation, which describes the quantum dynamics of a particle in the field of a one-dimensional delta potential (1DDP), are divided into two classes: some define pure states that have no free dynamics as t→∓∞t\to\mp\inftyt→∓∞; 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.
It is shown that in the state space of a quantum particle involved in a one-dimensional completed scattering (OCS) there is a superselection rule (SSR) induced by the dichotomous physical context that determines the properties of a quantum ensemble of scattering particles in the disjoint spatial regions located on the different sides of the potential barrier at asymptotically large distances from it. At the initial stage of the OCS this context is associated, in the general case, with two sources of particles located on the different sides of the barrier; and at the final stage it is associated, in the general case, with two particle detectors located on different sides of the barrier. The role of a superselection operator is played by the Pauli matrix σ3σ_3σ3​ that divides the space of in- and out-asymptotes into two coherent sectors. In the course of the OCS, its unitary quantum dynamics crosses these sectors. In particular, in the scattering problem with one source and two detectors, a pure initial state of a particle is converted into a mixed final state. According to this rule, the OCS is a complex quantum process consisting of two coherent subprocesses -- transmission and reflection; the average value of any observable as well as characteristic times can be defined for the subprocesses only. It is shown that the quantum dynamics of both subprocesses at all stages of scattering is uniquely determined by the initial state of the whole ensemble of particles and by the final states of transmitted and reflected particles.
It is shown that spontaneous symmetry breaking occurs in this scattering problem. As a result, the unitary, asymptotically free quantum dynamics of this process is unstable. It transforms into dynamics associated with the violation of PPP- and TTT-symmetries and the appearance of a superselection rule. Here, the non-unitary scattering state is a mixed vector state, which is a coherent superposition of transmission and reflection states related to different Hamiltonians. Superselection operators for the transmission and reflection subprocesses are defined.
A {\it completed} scattering of a particle on a static one-dimensional (1D) potential barrier is a combined quantum process to consist from two elementary sub-processes (transmission and reflection) evolved coherently at all stages of scattering and macroscopically distinct at the final stage. The existing model of the process is clearly inadequate to its nature: all one-particle "observables" and "tunneling times", introduced as quantities to be common for the sub-processes, cannot be experimentally measured and, consequently, have no physical meaning; on the contrary, quantities introduced for either sub-process have no basis, for the time evolution of either sub-process is unknown in this model. We show that the wave function to describe a completed scattering can be uniquely presented as the sum of two solutions to the Schrödinger equation, which describe separately the sub-processes at all stages of scattering. For symmetric potential barriers such solutions are found explicitly. For either sub-process we define the time spent, on the average, by a particle in the barrier region. We define it as the Larmor time. As it turned out, this time is just Buttiker's dwell time averaged over the corresponding localized state. Thus, firstly, we justify the known definition of the local dwell time introduced by Hauge and co-workers as well by Leavens and Aers, for now this time can be measured; secondly, we confirm that namely Buttiker's dwell time gives the energy-distribution for the tunneling time; thirdly, we state that all the definitions are valid only if they are based on the wave functions for transmission and reflection found in our paper. Besides, we define the exact and asymptotic group times to be auxiliary in timing the scattering process.