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Quelques aspects de la diffusion quantique : temps de retard, théorème de Levinson et potentiels dépendant du temps

Massimiliano Sassoli de Bianchi

Published Jan 1, 1995• 0 citations
Publisher

Abstract

In this Thesis we study several aspects of potential scattering in non-relativistic quantum mechanics. In the first part, we study the concept of time delay. More precisely, after an elementary introduction to the theory (chapter I), we clarify in chapter II the role played by the localizing regions in the definition of global time delay. In chapter III, we generalize the concept to arbitrary conditions of observation for the scattered particle (conditional time delay). In chapters IV and V, we address the problem of the measure of time delay by physical clocks and, more specifically, by the Larmor clock which exploit the mechanism of precession of a spin in a magnetic field. In the second part, we are interested in the spectral property of time delay, its connection with Levinson's theorem, and the application of the latter to one-dimensional systems. We derive Levinson's theorem in chapter VI, using as a single ingredient the completeness of physical states. In chapter VII, we apply Levinson's result to determine the number of bound-states of a finite periodic potential, as a function of its period. For this, we use the factorization property of the scattering matrix which we derive in chapter VIII in the more general situation of a particle with position-dependent mass. In the third part, we are concerned by time-dependent potentials. In chapter IX, we generalize the concept of time delay, and for the particular case of a periodic variation we derive a Levinson theorem. In chapter X, we consider potentials with very slow or very rapid variations in time. The low and high frequency limits are derived as well as their first corrections, and their physical significance discussed.

DOI
10.5075/epfl-thesis-1438

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•Jun 8, 2020•0 citations
In this Thesis we study several aspects of potential scattering in non-relativistic quantum mechanics. In the first part, we study the concept of time delay. More precisely, after an elementary introduction to the theory (chapter I), we clarify in chapter II the role played by the localizing regions in the definition of global time delay. In chapter III, we generalize the concept to arbitrary conditions of observation for the scattered particle (conditional time delay). In chapters IV and V, we address the problem of the measure of time delay by physical clocks and, more specifically, by the Larmor clock which exploit the mechanism of precession of a spin in a magnetic field. In the second part, we are interested in the spectral property of time delay, its connection with Levinson's theorem, and the application of the latter to one-dimensional systems. We derive Levinson's theorem in chapter VI, using as a single ingredient the completeness of physical states. In chapter VII, we apply Levinson's result to determine the number of bound-states of a finite periodic potential, as a function of its period. For this, we use the factorization property of the scattering matrix which we derive in chapter VIII in the more general situation of a particle with position-dependent mass. In the third part, we are concerned by time-dependent potentials. In chapter IX, we generalize the concept of time delay, and for the particular case of a periodic variation we derive a Levinson theorem. In chapter X, we consider potentials with very slow or very rapid variations in time. The low and high frequency limits are derived as well as their first corrections, and their physical significance discussed.
This paper states and proves Levinson's theorems for the classical scattering of two particles. The derivation is carried out by using time-delay theory. In both classical and quantum mechanics Levinson's theorems are shown to be the consequence of a common principle. This principle is the spectral property of time delay. Consider an arbitrary space region v. In quantum and classical mechanics a time-dependent scattering state traverses v in a definite time. The spectral property states that the sum of transit times for all orbits with total energy \ensuremath{\epsilon} is proportional to the state density in region v. In classical mechanics it is established that this property can be derived from simple features of the time-evolution group.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
Using the time-dependent scattering theory the authors prove that, in any spatial dimension and for arbitrary spin, the reading of the Larmor clock agrees with the global Eisenbud-Wigner time delay in the limit of an infinitesimal magnetic field. They show that convergence is also achieved at fixed energy (without oscillating terms) in the limit where the spatial switching on the field occurs on a much larger scale than the de Broglie wavelength of the particle. Finally, they investigate the functioning of the spin clock beyond the linear response regime.
This thesis concerns quantum states that are superpositions of discrete or of continuum energy levels. It develops mathematical methods to characterize the physical properties of these states and finds important consequences. Regarding rational extensions of the harmonic oscillator, no other authors had constructed the coherent states associated with the ladder operators of Marquette and Quesne. It was our aim to proceed with this construction, then to calculate physical properties of these coherent states, looking for classical or quantum behaviour. The method of partial wave analysis was known to be not applicable to the Coulomb potential for the scattering of plane waves. It was my aim to repeat the calculation for wavepacket scattering, to investigate whether the method became applicable. Other authors have constructed relativistic probability amplitudes and compared them to the scalar Klein-Gordon amplitude. In this context, it was my aim to question the conclusion of a minimum width for a massive particle. In the same context, I investigated the possibility of a four-vector probability current with the Newton-Wigner position probability density as the zero component. In Chapter 12, my aim was to construct a measure of localization for the photon, knowing from the work of other authors that a point-localized photon state is not possible.In Chapters 3, 4 and 5, our measures of classicality for the Barut-Girardello coherent states included plotting energy expectations as functions of the magnitude of the coherent state para- meter, z, position probability densities for the coherent states and the even and odd cat states, the Wigner functions for particular choices of z, the two-photon number probability distribu- tion and the linear entropy for a coherent state on a beamsplitter, position spreads to look for squeezing and the Mandel Q parameter. In Chapter 6, it was necessary to approximate an integral containing a particular Wigner rotation matrix. An approximation of general Wigner rotation matrices, at low angle, was derived and its precision evaluated numerically in Chapter 7. The exact values for the Coulomb phase shifts were used to construct the incoming wavefunction. Time reversal and a rotation were used to construct the outgoing wavefunction. The sums over l were evaluated numerically. In Chapter 10, four arguments were used against the minimum width. Comparison of the scalar amplitude and the relativistic probability amplitude for a Newton-Wigner localized state showed the root of the problem. A Lorentz transformation was applied to the scalar function to see if the “minimum” width became even smaller. A simple construction produced scalar functions with arbitrarily small widths. A relevant argument from Almeida and Jabs was included here. In Chapter 11, the algebra involving the four components of a four-vector current (at the origin) and the boost generators was derived. All of these commutators must be satisfied by a four-vector, providing a test for the claimed four-vector probability currents of other authors. In Chapter 12, field expectations were taken in a coherent state with a mean photon number of unity, built from identical single-photon states.In Chapters 3, 4 and 5, we found a mix of classical and quantum behaviour for the coherent states. In none of the systems studied did we see the high degree of classicality of the unmodified harmonic oscillator. The investigation of cat states was relevant to the fundamental problem of measurement in quantum mechanics. In Chapter 6, the use of wavepackets introduced a convergence factor into the sum over l, which diverges in the plane wave treatment. The elimination of a divergence is the most significant result of this thesis. Scattering probabilities remained finite at all angles, in disagreement with the diverging Rutherford formula, but only on small angular regions around forward scattering. Future work will investigate the measurability of the phenomena predicted here. In Chapter 10, all four arguments pointed to no minimum width for a massive particle. In Chapter 11, it was found that there can be no four-vector current density with the Newton-Wigner position probability density as the zero component, for any spin. Other authors were found to be in error when they claimed to have produced such a four-vector current density. This result strengthens the position of the Dirac current as the appropriate current for an electron. In Chapter 12, the coherent state field expectations were found to provide a meaningful measure of localization. In summary, wavepackets give a realism to the description of a scattering experiment that is absent with the use of plane waves. Phenomena not captured by the plane wave scattering amplitude, such as time delays, are calculable in the wavepacket context. Relativistic probability amplitudes provide a complete description of free particles, consistent with quantum mechanics and with special relativity. They predict agreement with the Heisenberg uncertainty principle and relativistic effects such as Lorentz contraction. The absence of relativistic probability amplitudes from standard textbooks on relativistic quantum mechanics is a serious omission.