On quantum mechanical decay processes

This thesis is concerned with quantum mechanical decay processes and their mathematical description. It consists out of three parts: In the first part we look at Laser induced ionization, whose mathematical description is often based on the so-called dipole approximation. Employing it essentially means to replace the Laser's vector potential $\vec A(\vec r,t)$ in the Hamiltonian by $\vec A(0, t).$ Heuristically this is justified under usual experimental conditions, because the Laser varies only slowly in $\vec r$ on atomic length scales. We make this heuristics rigorous by proving the dipole approximation in the limit in which the Laser's length scale becomes infinite compared to the atomic length scale. Our results apply to $N$-body Hamiltonians. In the second part we look at alpha decay as described by Skibsted (Comm. Math. Phys. 104, 1986) and show that Skibsted's model satisfies an energy-time uncertainty relation. Since there is no self-adjoint time operator, the uncertainty relation for energy and time can not be proven in the same way as the uncertainty relation for position and momentum. To define the time variance without a self-adjoint time operator, we will use the arrival time distribution obtained from the quantum current. Our proof of the energy-time uncertainty relation is then based on the quantitative scattering estimates that will be derived in the third part of the thesis and on a result from Skibsted. In addition to that, we will show that this uncertainty relation is different from the well known {\it linewidth-lifetime relation}. The third part is about quantitative scattering estimates, which are of interest in their own right. For rotationally symmetric potentials having support in $[0,R_V]$ we will show that for $R\geq R_V$, the time evolved wave function $e^{-iHt}\psi$ satisfies \begin{align}\nonumber \|\1_R e^{-iHt}\psi\|_2^2\leq c_1t^{-1}+c_2t^{-2}+c_3t^{-3}+c_4t^{-4} \end{align} with explicit quantitative bounds on the constants $c_n$ in terms of the resonances of the $S$-Matrix. While such bounds on \|\1_R e^{-iHt}\psi\|_2 have been proven before, the quantitative estimates on the constants $c_n$ are new. These results are based on a detailed analysis of the $S$-matrix in the complex momentum plane, which in turn becomes possible by expressing the $S$-matrix in terms of the Jost function that can be factorized in a Hadamard product.Gegenstand dieser Arbeit ist die mathematische Beschreibung von quantenmechanischen Zerfallsprozessen. Im ersten von drei Teilen, werden wir die durch Laser induzierte Ionisation von Atomen untersuchen, die üblicherweise mit Hilfe der sogenannten Dipolapproximation beschrieben wird. Bei dieser Approximation wird das Vektorpotential $\vec A(\vec r,t)$ des Lasers im Hamiltonoperator durch $\vec A(0, t)$ ersetzt, was oft dadurch gerechtfertigt ist, dass sich das Vektorpotential des Lasers auf atomaren Längenskalen in $\vec r$ kaum verändert. Ausgehend von dieser Heuristik werden wir die Dipolapproximation in dem Limes beweisen, in dem die Wellenlänge des Lasers im Verhältnis zur atomaren Längenskala unendlich groß wird. Unsere Resultate sind auf $N$-Teilchen Systeme anwendbar. Im zweiten Teil wenden wir uns dem Alphazerfallsmodell von Skibsted (Comm. Math. Phys. 104, 1986) zu und beweisen, dass es eine Energie-Zeit Unschärfe erfüllt. Da kein selbstadjungierter Zeitoperator existiert, kann die Energie-Zeit Unschärfe nicht auf gleiche Weise wie die Orts-Impuls Unschärfe bewiesen werden. Um ohne einen selbstadjungierten Zeitoperator Zugriff auf die Zeitvarianz zu bekommen, werden wir mit Hilfe des quantenmechanischen Wahrscheinlichkeitsstroms eine Ankunftszeitverteilung definieren. Der Beweis der Energie-Zeit Unschärfe folgt dann aus einem Resultat von Skibsted und aus den quantitativen Streuabschätzungen, die im dritten Teil der Dissertation bewiesen werden. Darüber hinaus werden wir zeigen, dass diese Unschärfe von der {\it linewidth-lifetime relation} zu unterscheiden ist. Hauptresultat des dritten Teils sind quantitative Streuabschätzungen, die als eigenständiges Resultat von Interesse sind. Für rotationssymmetrische Potentiale mit Träger in $[0,R_V]$ werden wir für alle $R\geq R_V$ die Abschätzung \begin{align}\nonumber \|\1_R e^{-iHt}\psi\|_2^2\leq c_1t^{-1}+c_2t^{-2}+c_3t^{-3}+c_4t^{-4} \end{align} beweisen und darüber hinaus, das ist das Novum, quantitative Schranken für die Konstanten $c_n$ angeben, die von den Resonanzen der $S$-Matrix abhängen. Um zu diesen Schranken zu gelangen, werden wir die analytische Struktur der $S$-Matrix studieren, indem wir die Beziehung der $S$-Matrix zur Jost-Funktion ausnutzen und die wiederum in ein Hadamard-Produkt zerlegen

On the Time-Dependent Analysis of Gamow Decay

Gamow's explanation of the exponential decay law uses complex "eigenvalues" and exponentially growing "eigenfunctions". This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wave functions. Observing that the time evolution of any wave function is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover the presentation can well be used in physics lectures to students.Comment: 10 pages, 4 figures; heuristic argument simplified, different example discussed, calculation of decay rate adde

Function and Assembly of a Chromatin-Associated RNase P that Is Required for Efficient Transcription by RNA Polymerase I

Background: Human RNase P has been initially described as a tRNA processing enzyme, consisting of H1 RNA and at least ten distinct protein subunits. Recent findings, however, indicate that this catalytic ribonucleoprotein is also required for transcription of small noncoding RNA genes by RNA polymerase III (Pol III). Notably, subunits of human RNase P are localized in the nucleolus, thus raising the possibility that this ribonucleoprotein complex is implicated in transcription of rRNA genes by Pol I. Methodology/Principal Findings: By using biochemical and reverse genetic means we show here that human RNase P is required for efficient transcription of rDNA by Pol I. Thus, inactivation of RNase P by targeting its protein subunits for destruction by RNA interference or its H1 RNA moiety for specific cleavage causes marked reduction in transcription of rDNA by Pol I. However, RNase P restores Pol I transcription in a defined reconstitution system. Nuclear run on assays reveal that inactivation of RNase P reduces the level of nascent transcription by Pol I, and more considerably that of Pol III. Moreover, RNase P copurifies and associates with components of Pol I and its transcription factors and binds to chromatin of the promoter and coding region of rDNA. Strikingly, RNase P detaches from transcriptionally inactive rDNA in mitosis and reassociates with it at G1 phase through a dynamic and stepwise assembly process that is correlated with renewal of transcription

Function and Assembly of a Chromatin-Associated RNase P that Is Required for Efficient Transcription by RNA Polymerase I

Human RNase P has been initially described as a tRNA processing enzyme, consisting of H1 RNA and at least ten distinct protein subunits. Recent findings, however, indicate that this catalytic ribonucleoprotein is also required for transcription of small noncoding RNA genes by RNA polymerase III (Pol III). Notably, subunits of human RNase P are localized in the nucleolus, thus raising the possibility that this ribonucleoprotein complex is implicated in transcription of rRNA genes by Pol I.By using biochemical and reverse genetic means we show here that human RNase P is required for efficient transcription of rDNA by Pol I. Thus, inactivation of RNase P by targeting its protein subunits for destruction by RNA interference or its H1 RNA moiety for specific cleavage causes marked reduction in transcription of rDNA by Pol I. However, RNase P restores Pol I transcription in a defined reconstitution system. Nuclear run on assays reveal that inactivation of RNase P reduces the level of nascent transcription by Pol I, and more considerably that of Pol III. Moreover, RNase P copurifies and associates with components of Pol I and its transcription factors and binds to chromatin of the promoter and coding region of rDNA. Strikingly, RNase P detaches from transcriptionally inactive rDNA in mitosis and reassociates with it at G1 phase through a dynamic and stepwise assembly process that is correlated with renewal of transcription.Our findings reveal that RNase P activates transcription of rDNA by Pol I through a novel assembly process and that this catalytic ribonucleoprotein determines the transcription output of Pol I and Pol III, two functionally coordinated transcription machineries

Ontogeny-Driven rDNA Rearrangement, Methylation, and Transcription, and Paternal Influence

Gene rearrangement occurs during development in some cell types and this genome dynamics is modulated by intrinsic and extrinsic factors, including growth stimulants and nutrients. This raises a possibility that such structural change in the genome and its subsequent epigenetic modifications may also take place during mammalian ontogeny, a process undergoing finely orchestrated cell division and differentiation. We tested this hypothesis by comparing single nucleotide polymorphism-defined haplotype frequencies and DNA methylation of the rDNA multicopy gene between two mouse ontogenic stages and among three adult tissues of individual mice. Possible influences to the genetic and epigenetic dynamics by paternal exposures were also examined for Cr(III) and acid saline extrinsic factors. Variables derived from litters, individuals, and duplicate assays in large mouse populations were examined using linear mixed-effects model. We report here that active rDNA rearrangement, represented by changes of haplotype frequencies, arises during ontogenic progression from day 8 embryos to 6-week adult mice as well as in different tissue lineages and is modifiable by paternal exposures. The rDNA methylation levels were also altered in concordance with this ontogenic progression and were associated with rDNA haplotypes. Sperm showed highest level of methylation, followed by lungs and livers, and preferentially selected haplotypes that are positively associated with methylation. Livers, maintaining lower levels of rDNA methylation compared with lungs, expressed more rRNA transcript. In vitro transcription demonstrated haplotype-dependent rRNA expression. Thus, the genome is also dynamic during mammalian ontogeny and its rearrangement may trigger epigenetic changes and subsequent transcriptional controls, that are further influenced by paternal exposures

Student Sessions

09:15 Peter Waller - Measuring algorithm timings in the ATLAS High Level Trigger 09:30 Robert Grummt - The LVL2-Trigger and its test run with cosmic radiation 09:45 Mathieu Aurousseau - From measurement to reality, analysis of photons in the ATLAS Combined Test Beam