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Progress of Theoretical Physics

A Model of the Cloud Chamber in Generalized Coherent State Formalism

K. Kakazu

Published Aug 1, 1990• 5 citations
Publisher

Abstract

Particle trajectory in the Wilson cloud chamber is explained by means of the perturbation theory and generalized coherent states. The model consists of an incoming particle, molecules and detectors (the immediate environment of the molecules). The particle and the molecules are treated as typical quantum systems, while each detector has many harmonic oscillators whose state are totally described by generalized coherent states. If a molecule is ionized through interaction with a particle, the ionized excited state is stable, that is, the molecule does not return to the ground state. The stability of excited states of the molecule is a result of properties of the generalized coherent states of the detector interacting with the molecule.

DOI
10.1143/ptp.84.195

Related papers

Stability of particle trajectories and generalized coherent states

Kiyotaka Kakazu, Shuichi Matsumoto

We show that particle trajectories in the Wilson cloud chamber are straight and stable, using perturbation theory and generalized coherent states. The model consists of an incoming particle, many molecules, and the immediate environment of each molecule acting as a detector. Each such detector has many harmonic oscillators whose states are totally described by generalized coherent states, because of which particle trajectories are almost stable. That is, if a molecule is ionized through its interaction with the particle, the ionized excited state does not return to the lower states. Many molecules can be ionized when the line joining them is practically parallel to the direction of motion of the particle. However, when the molecules do not lie on a straight line, all of them cannot be ionized. The particle trajectory that is made up of the ionized molecules, therefore, is straight.

Physical Review A•Nov 1, 1990•4 citations

The Spin–Boson Model

Abraham Nitzan

In a generic quantum mechanical description of a molecule interacting with its thermal environment, the molecule is represented as a few level system (in the simplest description just two, for example, ground and excited states) and the environment is often modeled as a bath of harmonic oscillators. The resulting theoretical framework is known as the spin–boson model, a term that seems to have emerged in the Kondo problem literature (which deals with the behavior of magnetic impurities in metals) during the 1960s, but is now used in a much broader context. Indeed, it has become one of the central models of theoretical physics, with applications in physics, chemistry, and biology that range far beyond the subject of this book. Transitions between molecular electronic states coupled to nuclear vibrations, environmental phonons, and photon modes of the radiation field fall within this class of problems. The present chapter discusses this model and some of its mathematical implications. The reader may note that some of the subjects discussed in Chapter 9 are reiterated here in this more general framework. In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field and provides good starting points for describing nuclear motions in molecules and in solid environments. It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach. In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models: Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property! We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9.

Oxford University Press eBooks•Apr 6, 2006•0 citations

On the Dynamics of Generalized Coherent States. II. Classical Equations of Motions

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Using the Klauder approach the stable evolution of generalized coherent states (GCS) for some groups (SU(2), SU(1,1) and SU(N)) is considered and it is shown that one and the same classical solution z(t) can correctly characterize the quantum evolution of many different (in general nonequivalent) systems. As examples some concrete systems are treated in greater detail: it is obtained that the nonstationary systems of the singular oscillator, of the particle motion in a magnetic field, and of the oscillator with friction all have stable SU(1,1) GCS whose quantum evolution is determined by one and the same classical function z(t). The physical properties of the constructed SU(1,1) GCS are discussed and it is shown particularly that in the case of discrete series D_k^+ they are those states for which the quantum mean values coincide with the statistical ones for an oscillator in a thermostat.

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Coherent atom–molecule oscillations with hyperspherical coordinates

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We investigate two-component boson systems consisting of atoms and binary bound molecules of the same atoms. We assume that a coherent superposition of atomic and molecular Bose?Einstein condensates is formed in a non-stationary state leading to oscillations between the two condensates. To facilitate the formulation, we first give analogies in neutrino oscillations applicable for two states of one particle. We then extend to many non-interacting particles in an external field treated in the mean-field approximation as for collective motion of finite systems. We then go beyond the mean field and provide a framework for interacting particles and correlated wavefunctions. The atom?molecule oscillation frequency is dominated by the molecular binding energy when the atom?atom scattering length is substantially smaller than the oscillator trap length. Comparable scattering lengths can completely change the oscillation frequency. The damping time of the oscillations decreases with atom number and scattering lengths, although a large increase can occur for special combinations of the different interactions.

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Joint excitation probability for two harmonic oscillators in dimension one and the Mott problem

Gianfausto Dell’Antonio, Rodolfo Figari, Alessandro Teta

We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions $a_1$, $a_2$, with $a_1 >0$, $|a_2|>a_1$, in the two possible situations: $a_2>0$ and $a_2 <0$. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %$\pm M v_0$. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability $\mathcal{P}^{-}_{n_1 n_2} (t)$ (resp. $\mathcal{P}^{+}_{n_1 n_2} (t)$) that both oscillators are in the excited states labelled by $n_1$, $n_2 >0$ at time $t > |a_2| v_0^{-1}$ when $a_2 <0$ (resp. $a_2 >0$). We prove that $\mathcal{P}_{n_1 n_2}^- (t)$ is negligible with respect to $\mathcal{P}_{n_1 n_2}^+ (t)$, up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate.

arXiv:0711.2230•Nov 14, 2007•1 citation

Generalized coherent states

T. G. Philbin

In the coherent state of the harmonic oscillator, the probability density is that of the ground state subjected to an oscillation along a classical trajectory. Senitzky and others pointed out that there are states of the harmonic oscillator corresponding to an identical oscillatory displacement of the probability density of any energy eigenstate. These generalizations of the coherent state are rarely discussed, yet they furnish an interesting set of quantum states of light that combine features of number states and coherent states. Here, we give an elementary account of the quantum optics of generalized coherent states.

arXiv:1311.1920•American Journal of Physics•Jul 23, 2014•48 citations
Particle trajectory in the Wilson cloud chamber is explained by means of the perturbation theory and generalized coherent states. The model consists of an incoming particle, molecules and detectors (the immediate environment of the molecules). The particle and the molecules are treated as typical quantum systems, while each detector has many harmonic oscillators whose state are totally described by generalized coherent states. If a molecule is ionized through interaction with a particle, the ionized excited state is stable, that is, the molecule does not return to the ground state. The stability of excited states of the molecule is a result of properties of the generalized coherent states of the detector interacting with the molecule.
We show that particle trajectories in the Wilson cloud chamber are straight and stable, using perturbation theory and generalized coherent states. The model consists of an incoming particle, many molecules, and the immediate environment of each molecule acting as a detector. Each such detector has many harmonic oscillators whose states are totally described by generalized coherent states, because of which particle trajectories are almost stable. That is, if a molecule is ionized through its interaction with the particle, the ionized excited state does not return to the lower states. Many molecules can be ionized when the line joining them is practically parallel to the direction of motion of the particle. However, when the molecules do not lie on a straight line, all of them cannot be ionized. The particle trajectory that is made up of the ionized molecules, therefore, is straight.
In a generic quantum mechanical description of a molecule interacting with its thermal environment, the molecule is represented as a few level system (in the simplest description just two, for example, ground and excited states) and the environment is often modeled as a bath of harmonic oscillators. The resulting theoretical framework is known as the spin–boson model, a term that seems to have emerged in the Kondo problem literature (which deals with the behavior of magnetic impurities in metals) during the 1960s, but is now used in a much broader context. Indeed, it has become one of the central models of theoretical physics, with applications in physics, chemistry, and biology that range far beyond the subject of this book. Transitions between molecular electronic states coupled to nuclear vibrations, environmental phonons, and photon modes of the radiation field fall within this class of problems. The present chapter discusses this model and some of its mathematical implications. The reader may note that some of the subjects discussed in Chapter 9 are reiterated here in this more general framework. In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field and provides good starting points for describing nuclear motions in molecules and in solid environments. It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach. In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models: Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property! We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9.
Using the Klauder approach the stable evolution of generalized coherent states (GCS) for some groups (SU(2), SU(1,1) and SU(N)) is considered and it is shown that one and the same classical solution z(t) can correctly characterize the quantum evolution of many different (in general nonequivalent) systems. As examples some concrete systems are treated in greater detail: it is obtained that the nonstationary systems of the singular oscillator, of the particle motion in a magnetic field, and of the oscillator with friction all have stable SU(1,1) GCS whose quantum evolution is determined by one and the same classical function z(t). The physical properties of the constructed SU(1,1) GCS are discussed and it is shown particularly that in the case of discrete series D_k^+ they are those states for which the quantum mean values coincide with the statistical ones for an oscillator in a thermostat.
We investigate two-component boson systems consisting of atoms and binary bound molecules of the same atoms. We assume that a coherent superposition of atomic and molecular Bose?Einstein condensates is formed in a non-stationary state leading to oscillations between the two condensates. To facilitate the formulation, we first give analogies in neutrino oscillations applicable for two states of one particle. We then extend to many non-interacting particles in an external field treated in the mean-field approximation as for collective motion of finite systems. We then go beyond the mean field and provide a framework for interacting particles and correlated wavefunctions. The atom?molecule oscillation frequency is dominated by the molecular binding energy when the atom?atom scattering length is substantially smaller than the oscillator trap length. Comparable scattering lengths can completely change the oscillation frequency. The damping time of the oscillations decreases with atom number and scattering lengths, although a large increase can occur for special combinations of the different interactions.
We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions a1a_1a1​, a2a_2a2​, with a1>0a_1 >0a1​>0, ∣a2∣>a1|a_2|>a_1∣a2​∣>a1​, in the two possible situations: a2>0a_2>0a2​>0 and a2<0a_2 <0a2​<0. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %±Mv0\pm M v_0±Mv0​. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability Pn1n2−(t)\mathcal{P}^{-}_{n_1 n_2} (t)Pn1​n2​−​(t) (resp. Pn1n2+(t)\mathcal{P}^{+}_{n_1 n_2} (t)Pn1​n2​+​(t)) that both oscillators are in the excited states labelled by n1n_1n1​, n2>0n_2 >0n2​>0 at time t>∣a2∣v0−1t > |a_2| v_0^{-1}t>∣a2​∣v0−1​ when a2<0a_2 <0a2​<0 (resp. a2>0a_2 >0a2​>0). We prove that Pn1n2−(t)\mathcal{P}_{n_1 n_2}^- (t)Pn1​n2​−​(t) is negligible with respect to Pn1n2+(t)\mathcal{P}_{n_1 n_2}^+ (t)Pn1​n2​+​(t), up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate.
In the coherent state of the harmonic oscillator, the probability density is that of the ground state subjected to an oscillation along a classical trajectory. Senitzky and others pointed out that there are states of the harmonic oscillator corresponding to an identical oscillatory displacement of the probability density of any energy eigenstate. These generalizations of the coherent state are rarely discussed, yet they furnish an interesting set of quantum states of light that combine features of number states and coherent states. Here, we give an elementary account of the quantum optics of generalized coherent states.