We consider a general reaction process in which two entrance-channel particles, the beam and target particles, interact off-resonance and produce two exit-channel particles. In the c.m. frame we derive a general expression relating the probability density of the exit-channel particles to that of the entrance-channel particles. Furthermore, we show that whenever the coherence length, ${w}_{B}$ , of the beam particle is much larger than the coherence length, ${w}_{T}$ , of the target particle, the wave function of each exit-channel particle may be expressed explicitly in terms of the wave function of the beam particle. We also derive expressions for the coherence time and coherence length of each exit-channel particle. In addition, whenever the speed of the target particle is much less than the speed ${v}_{i}$ (i=1,2) of an exit-channel particle, we show that a simple relationship exists between the coherence length, ${w}_{\mathrm{ci}}$ , of the exit-channel particle and the beam particle: ${w}_{\mathrm{ci}}$ = ${v}_{i}$ ( ${w}_{B}$ /\ensuremath{\Vert} ${\mathrm{v}}_{B}$ ${\mathrm{\ensuremath{-}}\mathrm{v}}_{T}$\ensuremath{\Vert}){1 ${\mathrm{\ensuremath{-}}[(\mathrm{v}}_{\mathrm{T}}$\ensuremath{\cdot}e${\mathrm{^}}_{3}$)/ ${v}_{i}$ ](costheta')}, where the velocity of the beam particle is ${\mathrm{v}}_{B}$ ${(\mathrm{v}}_{B}$ = ${v}_{B}$ e${^}_{3}$), and the velocity of the target particle is ${\mathrm{v}}_{T}$ ${(\mathrm{v}}_{T}$ = ${v}_{T}$ e${^}_{3}$). The angle theta' is the spherical polar angular coordinate, measured with respect to the unit vector e${^}_{3}$, of the point at which the coherence length ${w}_{\mathrm{ci}}$ is to be determined. The above relationship is applied in the case of photoionization of the hydrogen atom at low energies to obtain realistic estimates for the coherence lengths of the ionized electron and proton. The coherence length of the electron is shown to be of nearly macroscopic dimensions.We consider a general reaction process in which two entrance-channel particles, the beam and target particles, interact off-resonance and produce two exit-channel particles. In the c.m. frame we derive a general expression relating the probability density of the exit-channel particles to that of the entrance-channel particles. Furthermore, we show that whenever the coherence length, ${w}_{B}$, of the beam particle is much larger than the coherence length, ${w}_{T}$, of the target particle, the wave function of each exit-channel particle may be expressed explicitly in terms of the wave function of the beam particle. We also derive expressions for the coherence time and coherence length of each exit-channel particle. In addition, whenever the speed of the target particle is much less than the speed ${v}_{i}$ (i=1,2) of an exit-channel particle, we show that a simple relationship exists between the coherence length, ${w}_{\mathrm{ci}}$, of the exit-channel particle and the beam particle: ${w}_{\mathrm{ci}}$=${v}_{i}$(${w}_{B}$/\ensuremath{\Vert}${\mathrm{v}}_{B}$${\mathrm{\ensuremath{-}}\mathrm{v}}_{T}$\ensuremath{\Vert}){1 ${\mathrm{\ensuremath{-}}[(\mathrm{v}}_{\mathrm{T}}$\ensuremath{\cdot}e${\mathrm{^}}_{3}$)/ ${v}_{i}$](costheta')}, where the velocity of the beam particle is ${\mathrm{v}}_{B}$ ${(\mathrm{v}}_{B}$=${v}_{B}$e${^}_{3}$), and the velocity of the target particle is ${\mathrm{v}}_{T}$ ${(\mathrm{v}}_{T}$=${v}_{T}$e${^}_{3}$). The angle theta' is the spherical polar angular coordinate, measured with respect to the unit vector e${^}_{3}$, of the point at which the coherence length ${w}_{\mathrm{ci}}$ is to be determined. The above relationship is applied in the case of photoionization of the hydrogen atom at low energies to obtain realistic estimates for the coherence lengths of the ionized electron and proton. The coherence length of the electron is shown to be of nearly macroscopic dimensions.

Physical Review AJan 1, 19860 citations

Coherent States

WORLD SCIENTIFIC eBooksMar 1, 20040 citations

Coherent States

P. L. Knight, L. Allen

Elsevier eBooksJan 1, 19830 citations

Wave‐Particle‐Mixedness Relationships Based on l p ‐norm Coherence

Weining Li, Ming Fang, Yuan-Hong Tao +2 more

ABSTRACT We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the ‐norm coherence. By conforming that the ‐norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal ‐norm coherence and the general ‐norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For , we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the ‐norm coherence given by the ‐norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.ABSTRACT We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the ‐norm coherence. By conforming that the ‐norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal ‐norm coherence and the general ‐norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For , we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the ‐norm coherence given by the ‐norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.

Annalen der PhysikFeb 28, 20261 citation

Wave-Particle-Mixedness Relationships Based on lp-norm Coherence

Wei-Ning Li, Ming Fang, Yuan-Hong Tao +2 more

We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the lp-norm coherence. By conforming that the lp-norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal l2-norm coherence and the general l2-norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For 1<= p <2, we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the lp-norm coherence given by the l2-norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the lp-norm coherence. By conforming that the lp-norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal l2-norm coherence and the general l2-norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For 1<= p <2, we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the lp-norm coherence given by the l2-norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.

quant-pharXiv:2511.05937Nov 11, 20250 citations

Coherent States

WORLD SCIENTIFIC eBooksMar 1, 20060 citations

{w}_{B}

, of the beam particle is much larger than the coherence length,

{w}_{T}

, of the target particle, the wave function of each exit-channel particle may be expressed explicitly in terms of the wave function of the beam particle. We also derive expressions for the coherence time and coherence length of each exit-channel particle. In addition, whenever the speed of the target particle is much less than the speed

{v}_{i}

(i=1,2) of an exit-channel particle, we show that a simple relationship exists between the coherence length,

{w}_{\mathrm{ci}}

, of the exit-channel particle and the beam particle:

{w}_{\mathrm{ci}}

{v}_{i}

(

{w}_{B}

/\ensuremath{\Vert}

{\mathrm{v}}_{B}

${\mathrm{\ensuremath{-}}\mathrm{v}}_{T}$\ensuremath{\Vert}){1 ${\mathrm{\ensuremath{-}}[(\mathrm{v}}_{\mathrm{T}}$\ensuremath{\cdot}e${\mathrm{^}}_{3}$)/

{v}_{i}

](costheta')}, where the velocity of the beam particle is

{\mathrm{v}}_{B}

{(\mathrm{v}}_{B}

{v}_{B}

e${^}_{3}$), and the velocity of the target particle is

{\mathrm{v}}_{T}

{(\mathrm{v}}_{T}

{v}_{T}

e${^}_{3}$). The angle theta' is the spherical polar angular coordinate, measured with respect to the unit vector e${^}_{3}$, of the point at which the coherence length

{w}_{\mathrm{ci}}

is to be determined. The above relationship is applied in the case of photoionization of the hydrogen atom at low energies to obtain realistic estimates for the coherence lengths of the ionized electron and proton. The coherence length of the electron is shown to be of nearly macroscopic dimensions.