Bulletin of the Russian Academy of Sciences Physics

A self-consistent description of particle scattering on a bound system that conserves unitarity

Н. Ф. Голованова

Published Jul 1, 2015• 0 citations

Abstract

A new way of solving the problem of scattering while conserving unitarity for real and complex interaction potentials is proposed: Finding a self-consistent solution to the many-particle Schrödinger equation.

DOI: 10.3103/s1062873815040115

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Bulletin of the Russian Academy of Sciences Physics

A self-consistent description of particle scattering on a bound system that conserves unitarity

Н. Ф. Голованова

Published Jul 1, 2015• 0 citations

Publisher

Abstract

DOI: 10.3103/s1062873815040115

Related papers

Simple applications

Peter Holland

Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]Stationary states General properties We shall consider first some general characteristics of the ‘stationary states’. These are eigenfunctions of the Hamiltonian operator: The requirement that ψ satisfies the Schrödinger equation fixes its time dependence: Notice that (4.1.2) can only be a solution if the external potential V is independent of the time: V = V ( x ). The initial function ψ 0 is not arbitrary but must satisfy the time-independent Schrödinger equation (4.1.1). Writing ψ = R e iS /ħ one thus has The following deductions may be made from (4.1.3): ( a ) The probability density is independent of the time: |ψ| 2 = R 2 0 ( X ). ( b ) The quantum potential Q = (- ħ2/2 mR 0 )∇ 2 R 0 is time-independent. Therefore so is the total effective potential: ( c ) The velocity field v = (l/ m )∇ S 0 is independent of time and hence the trajectories, found by solving m = v ( x )| x = x ( t ) , are solutions of Moreover, ψ 0 ( x ) is often a real function for many bound state problems of interest (although not for scattering states) which implies that the velocity is zero. The particle is at rest where one would classically expect it to move since the quantum force (- ∇ Q ) cancels the classical force (- ∇ K ). ( d ) The energy of all particles in the ensemble, -∂ S /∂ t , is a constant of the motion and equal to the energy eigenvalue: with (4.1.4), we have an example of the conservation of energy (cf. §3.9.3). ( e ) The total field energy is conserved: 〈Ĥ〉 = E . […]

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We develop a new method of constructing a large N asymptotic series in powers of $N^{-1/2}$ for the function of N arguments which is a solution to the Cauchy problem for the equation of a special type. Many-particle Wigner, Schrödinger and Liouville equations for a system of a large number of particles are of this type, when the external potential is of order O(1), while the coefficient of the particle interaction potential is 1/N; the potentials can be arbitrary smooth bounded functions. We apply this method to equations for N-particle states corresponding to the N-th tensor power of an abstract Hamiltonian algebra of observables. In particular, we show for the case of multiparticle Schrödinger-like equations that the property of N-particle wave function to be approximately equal at large N to the product of one-particle wave functions does not conserve under time evolution, while the same property for the correlation functions of the finite order is known to conserve(such hypothesis being the quantum analog of the chaos conservation hypothesis put forward by M.Kac in 1956 was proved by the analysis of the BBGKY-like hierarchy of equations). In order to find a leading asymptotics for the N-particle wave function, one should use not only the solution to the well- known Hartree equation being derivable from the BBGKY approach but also the solution to another (Riccati-type) equation presented in this paper. We also consider another interesting case when one adds to the N-particle system under consideration one more particle interacting with the system with the coefficient of the interaction potential of order O(1).It happens that in this case one should investigate not a single Hartree-like equation but a set of such equations, and the chaos will not conserve even for the correlators.We develop a new method of constructing a large N asymptotic series in powers of $N^{-1/2}$ for the function of N arguments which is a solution to the Cauchy problem for the equation of a special type. Many-particle Wigner, Schrödinger and Liouville equations for a system of a large number of particles are of this type, when the external potential is of order O(1), while the coefficient of the particle interaction potential is 1/N; the potentials can be arbitrary smooth bounded functions. We apply this method to equations for N-particle states corresponding to the N-th tensor power of an abstract Hamiltonian algebra of observables. In particular, we show for the case of multiparticle Schrödinger-like equations that the property of N-particle wave function to be approximately equal at large N to the product of one-particle wave functions does not conserve under time evolution, while the same property for the correlation functions of the finite order is known to conserve(such hypothesis being the quantum analog of the chaos conservation hypothesis put forward by M.Kac in 1956 was proved by the analysis of the BBGKY-like hierarchy of equations). In order to find a leading asymptotics for the N-particle wave function, one should use not only the solution to the well- known Hartree equation being derivable from the BBGKY approach but also the solution to another (Riccati-type) equation presented in this paper. We also consider another interesting case when one adds to the N-particle system under consideration one more particle interacting with the system with the coefficient of the interaction potential of order O(1).It happens that in this case one should investigate not a single Hartree-like equation but a set of such equations, and the chaos will not conserve even for the correlators.

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In the standard Breit-Wigner approach to scattering the phase shift is to have a form $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ at a real energy resonance. This leads to complex energy poles in the scattering amplitude at $E_{\rm BW}=E_1-iΓ_1$ , poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ is not always a good description of the real energy scattering amplitude, that $Γ_1$ can be negative, that $E_{\rm BW}$ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear $PT$ symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs $E_{\mp}=E_2\mp i Γ_2$ with $E_- \neq E_{\rm BW}$ , doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.In the standard Breit-Wigner approach to scattering the phase shift is to have a form $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ at a real energy resonance. This leads to complex energy poles in the scattering amplitude at $E_{\rm BW}=E_1-iΓ_1$, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ is not always a good description of the real energy scattering amplitude, that $Γ_1$ can be negative, that $E_{\rm BW}$ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear $PT$ symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs $E_{\mp}=E_2\mp i Γ_2$ with $E_- \neq E_{\rm BW}$, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.

arXiv:2605.28756May 27, 20260 citations

We develop a new method of constructing a large N asymptotic series in powers of

N^{-1/2}

for the function of N arguments which is a solution to the Cauchy problem for the equation of a special type. Many-particle Wigner, Schrödinger and Liouville equations for a system of a large number of particles are of this type, when the external potential is of order O(1), while the coefficient of the particle interaction potential is 1/N; the potentials can be arbitrary smooth bounded functions. We apply this method to equations for N-particle states corresponding to the N-th tensor power of an abstract Hamiltonian algebra of observables. In particular, we show for the case of multiparticle Schrödinger-like equations that the property of N-particle wave function to be approximately equal at large N to the product of one-particle wave functions does not conserve under time evolution, while the same property for the correlation functions of the finite order is known to conserve(such hypothesis being the quantum analog of the chaos conservation hypothesis put forward by M.Kac in 1956 was proved by the analysis of the BBGKY-like hierarchy of equations). In order to find a leading asymptotics for the N-particle wave function, one should use not only the solution to the well- known Hartree equation being derivable from the BBGKY approach but also the solution to another (Riccati-type) equation presented in this paper. We also consider another interesting case when one adds to the N-particle system under consideration one more particle interacting with the system with the coefficient of the interaction potential of order O(1).It happens that in this case one should investigate not a single Hartree-like equation but a set of such equations, and the chaos will not conserve even for the correlators.

In the standard Breit-Wigner approach to scattering the phase shift is to have a form

\tanδ_{\rm BW} =Γ_1/(E_1-E)

at a real energy resonance. This leads to complex energy poles in the scattering amplitude at

E_{\rm BW}=E_1-iΓ_1

, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting

\tanδ_{\rm BW} =Γ_1/(E_1-E)

is not always a good description of the real energy scattering amplitude, that

Γ_1

can be negative, that

E_{\rm BW}

is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear

PT

symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs

E_{\mp}=E_2\mp i Γ_2

with

E_- \neq E_{\rm BW}

, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.