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This work reports an effort to introduce a few advanced concepts in quantum physics to undergraduate students who have only a minimal familiarity with wave mechanics. The geometric phase is not introduced in most introductory courses on quantum mechanics. It is of pivotal significance for the understanding of topological phase transitions which are becoming increasingly important in a large number of physical phenomena, including those that involve platforms for robust quantum computing. This article would provide a gentle introduction to the geometric phase. Toward this goal, Feynman’s path integral approach to quantum mechanics and its application in the interpretation of the wondrous Aharonov–Bohm effect is introduced in a gentle manner. A computer simulation of this interesting phenomenology is presented to help gain an early insight into some advanced concepts.

A theoretical study of the dipole photoelectron angular distribution asymmetry parameter, β, of the 6s, 5s and 4s subshells has been conducted for a range of closed-shell heavy, Hg (Z = 80), Rn (Z = 86), Ra (Z = 88) and superheavy, No (Z = 102), Cn (Z = 112), Og (Z = 118), Ubn (Z = 120) elements to understand the deviation of β from the nonrelativistic value of 2 owing to relativistic effects. It was found that the deviation, strongest in Cooper minimum regions, persists over all energy for all ns subshells, and the deviations increase with Z. Three levels of relativistic calculations were performed in order to pinpoint the particular interactions responsible for the details of the behavior of the ns β’s as functions of energy, n, and Z and, in addition to the effects of the (relativistic) spin–orbit interaction, interchannel coupling was found to be of crucial importance in the determination of the β parameter in most of the situations studied.

If at first the idea is not absurd, then there is no hope for it. —Albert Einstein GEOMETRY OF THE SPACE–TIME CONTINUUM The gravitational interaction is the earliest physical interaction that humans have registered. The earliest speculations about just what is the nature of gravity were not merely wrong, but absurdly far-fetched. Ancient philosophers even conjectured that the earth is the natural abode of things, and objects fall down when they are dropped just as horses return to their stables. Various theories of gravity were proposed, and the one that lasted much is that developed by Isaac Newton in the seventeenth century. Newton's work on gravity integrated the dynamics of astronomical objects with that of falling apples or coconuts, determined by one common principle (Chapter 8). We celebrate this principle as Newton's one-over-distance-square law of gravity. An amazing consequence of the constancy of the speed of light in all inertial frames of reference that we studied in the previous chapter is the time-dilation and Lorentz contraction (also called the length contraction ). The phenomenon that is responsible for the traveling twin to age slower than the home-bound twin holds for any and every object in motion. We have already noted that this happens to decaying muons. Essentially, the faster you move through space, the slower you move through time, in the spacetime continuum. We all enjoy raising our speed, covering more distance in lesser, and lesser, time. Let us therefore ask, to what extent can we speed up an object? We ask if there is a natural limit for this. If you look back into the relations for time-dilation and the length contraction in the previous chapter, you will recognize that if v = c , the effect of time-dilation would be such that the traveling twin will simply stop ageing. Time would stop for her; time freezes. The effect of Lorentz contraction would also be total; she would think that the rest of the universe has spatially contracted to a point. She is therefore already everywhere (along the line of motion). All of these dramatic aftermaths are because of a simple fundamental property that the speed of the headlight of a car coming toward you at a velocity v is no different from that of the tail light of another that is receding away from you.

When I was in high school, my Physics teacher—whose name was Mr Bader—called me down one day after physics class and said, ‘You look bored; I want to tell you something interesting’. Then he told me something which I found absolutely fascinating, and have since then, always found fascinating. Every time the subject comes up, I work on it… The subject is this—the principle of least action. —Richard P. Feynman THE VARIATIONAL PRINCIPLE AND EULER–LAGRANGE'S EQUATION OF MOTION In the previous chapters, we have worked with the Newtonian formulation of classical mechanics. Its central theme relies on the use of ‘force’ as the very cause of change in momentum. The cornerstone of Newtonian mechanics is this principle of causality. It is expressed in Newton's second law as a linear relation between the acceleration and the force. It is the result of the equality between the force and the rate of change of momentum. The relation between force and momentum is at the very heart of Newtonian formulation of classical mechanics. It turns out that classical mechanics has an alternative but equivalent formulation, based on what is known as the ‘ variational principle ’, or ‘ Hamilton's principle of variation ’. In many universities, the principle of variation [1, 2] is introduced after a few years of college education in physics, and after a few courses on mechanics, including electrodynamics. However, there have been a few proposals [3, 4, 5] which recommend an early exposure in college curriculum to this fascinating approach. In fact, Richard Feynman was introduced to the principle of variation by his high school teacher, Mr Bader. Feynman went on to develop the path integral approach to the quantum theory based on the principle of variation. The path integral approach to quantum mechanics provides an alternative formulation of the quantum theory; it is equivalent to Heisenberg's uncertainty principle, and the Schrodinger equation. It has the capacity to describe a mechanical system and to account for how it evolves with time. The variational principle can be adapted to provide a backward integration of classical mechanics as an approximation toward the development of quantum theory. Newtonian formulation is not suitable for this purpose.

Lost time is never found again. —Benjamin Franklin DISSIPATIVE SYSTEMS Energy dissipative systems prompt us to think of physical phenomena in which energy is not conserved, it is lost. We attribute these losses to ‘friction’, which is the common term used to describe energy dissipation. Now, in our everyday experience, we are primarily involved with the gravitational and electromagnetic interactions, and both of these are essentially conservative. What is it, then, that makes friction non-conservative ? What does non-conservation of energy, or ‘energy-loss’, really mean? Fundamental interactions in nature allow energy to be changed from one form to another, but not created or destroyed. It therefore seems that the term ‘energy loss’ is used somewhat loosely. We must be really careful when we talk about dissipative phenomena. Losing money is often a matter of concern, as also other things we sometimes ‘lose’ from time to time, including time itself. Time wasted does not ever come back, of course; but nor does the time well-spent. The difference between the two is that the latter is accounted for by the gains made, and for the former there is simply no account. Isn't it merely a matter of book-keeping? It is not at all uncommon that we plan to do something during the day, and end up not doing it. We then sit and wonder where lost track of time. Did the day just skip over the afternoon and ring in the evening? If that did not happen, where was the time ‘lost’? You possibly remember everything that you did since morning, and you may be able to account for every hour you spent, except perhaps for what happened between 2:30 pm and 3 pm. That was the time you ransacked your house to find your mathematics text book, and did not realize that it took you half an hour to find it. The book had not really vanished, even if you suspected that it was lost. You had to look for it all over, from your Dad's room to your Sister's. As such, neither the book was lost, nor was half an hour scooped out of the day.

I am convinced that chaos research will bring about a revolution in natural sciences similar to that produced by quantum mechanics. — Gerd Binnig LEARNING FROM NUMBERS The equations of motion of classical mechanics, whether Newton's, Lagrange's, or Hamilton's, have us believe that given the state of the system at a particular time, one can always predict what its state would be any time later, or, for that matter, what it was any time earlier. This is because of the fact that the equations of motion are symmetric with respect to time-reversal: ( t )→(– t ). Classical mechanics relies on the assumption that position q and momentum p of a mechanical system are simultaneously knowable. Together, the pair ( q , p ) provides a signature of the state of the system . Their time-dependence, i.e., provided by the equations of motion, accurately describes their temporal evolution. For macroscopic objects, this is an excellent approximation, and the classical laws of mechanics are stringently deterministic. This is stringently correct, but there is an important caveat, expressed succinctly by Stephen Hawking: “ Our ability to predict the future is severely limited by the complexity of the equations, and the fact that they often have a property called chaos… a tiny disturbance in one place, can cause a major change in another. ” The difficulty Hawking alludes to has nothing to do with the quantum principle of uncertainty, but to a challenge within the framework of the fully deterministic classical theory. The solution to the temporal evolution may become chaotic , even as they remain deterministic, due to extreme sensitivity to the initial conditions that are necessary to obtain the solution to the equation of motion. Careful admission of the previous remark would prepare you to embark your journey on the exciting field of chaos. Along the way, you will also meet objects having weird fractional dimensions. The field covered by chaos theory is vast, though relatively young. It is a very rich field and can be introduced from a variety of perspectives. The general field of chaos theory is often regarded as a study of a ‘dynamical system’ which is just about any quantity which changes, and one has reasons to track these changes and the sequence of values it may take, for example, over a time interval.

There is no greater burden than an unfulfilled potential. —Charles M. Schulz THE SCALAR FIELD, DIRECTIONAL DERIVATIVE, AND GRADIENT In the discussion on Fig. 2.4 (Chapter 2), we learned that it was neither innocuous to define a vector merely as a quantity that has both direction and magnitude, nor to define a scalar simply as a quantity that has magnitude alone. It is not that the properties referred here of a scalar and a vector are invalid. Rather, it is to be understood that these properties do not provide an unambiguous definition . Only a signature criterion of a physical quantity can be used to define it. We therefore introduced, in Chapter 2, comprehensive definitions of the scalar as a tensor of rank 0, and of the vector as a tensor of rank 1. In this chapter we shall acquaint ourselves with the mathematical framework in which the laws of fluid mechanics and electrodynamics are formulated using vector algebra and vector calculus. In fact, the techniques are used not merely in these two important branches of classical mechanics, but also in very various other subdivisions of physics. The background material seems at times to be intensely mathematical, but that is only because the laws of nature engage a mathematical formulation very intimately, as we encounter repeatedly in the analysis of physical phenomena. There are many excellent books in college libraries from which one can master the mathematical methods. These topics are extremely enjoyable to learn; they help us develop rigorous insights in the laws of nature. The literature on these topics is vast. A couple of illustrative books [1, 2] are suggested for further reading. We consider the example of a particular scalar function, namely the temperature distribution in a room. The temperature is a physical property at a particular point in space, such as the point P in Fig. 10.1. Depending on the distribution of the sources of heat in the region that surrounds the point P , temperature may be different from point to point in space, and also possibly from time to time. The reason the temperature at a point is a scalar, is that its value at that point is independent of where the observer's frame of reference is located, and also independent of how it is oriented.

In this proceeding, we present a study of the spin–orbit interaction dominated features in time domain photoelectron dynamics. In particular, a case study of Wigner–Eisenbud–Smith (WES) time delay in the photoionization of Xe 4d subshell in the near-threshold region is presented and discussed. We show that the Spin–Orbit Interaction Activated Interchannel Coupling (SOIAIC) produces a prominent resonance-like structure in the WES time delay spectrum.

Synopsis Calculations of the photoionization cross sections for the superheavy atoms of atomic numbers 102, 112 and 118 have been performed using the relativistic-random-phase approximation (RRPA) which is based on the Dirac equation. The results show that relativistic effects cause anomalies in level ordering for some inner shells, and that interchannel coupling is of great importance.

The results of calculations of the atomic photoionisation cross sections, angular distribution asymmetry parameters, branching ratios and spin polarisation parameters for the 4d, 4p, 4s, 3d and 3p subshells of palladium are reported. Calculations have been performed for a photon energy range of about 0.34-28.0 au. The positions of the Cooper minima in the 4d and 4p photoionisation cross sections are located in the continuum. Also, the study shows the 4s and 3p Cooper minima to be in the discrete part of the spectrum. The centrifugal barrier faced by the continuum f electrons causes a 'shape resonance' in the 4d and 3d cross sections and also affects the branching ratio for the sigma (nd5/2): sigma (nd3/2) cross sections. The branching ratio in the 4d cross section is influenced by the fact that the Cooper minima in the individual relativistic dipole, 4d5/2,3/2 to in f7/2,3/2 channels occur at different photon energies. Threshold effects and dynamical factors such as the occurrence of Cooper minima have a marked effect on the energy dependence of the spin polarisation parameters.

Two decades ago, it was predicted [Y. S. Kim et al., Phys. Rev. Lett. 46, 1326 (1981)] that relativistic effects should alter the dynamics of the photoionization process in the vicinity of Cooper minima. The present experimental and theoretical study of the angular distributions of Xe 4d(3/2) and 4d(5/2) photoelectrons demonstrates this effect for the first time. The results clearly imply that relativistic effects are likely to be important for intermediate- Z atoms at most energies.

Two decades ago, it was predicted [Y. S. Kim et al., Phys. Rev. Lett. 46, 1326 (1981)] that relativistic effects should alter the dynamics of the photoionization process in the vicinity of Cooper minima. The present experimental and theoretical study of the angular distributions of Xe 4d(3/2) and 4d(5/2) photoelectrons demonstrates this effect for the first time. The results clearly imply that relativistic effects are likely to be important for intermediate- Z atoms at most energies.

Near the ionisation threshold, the behaviour of the generalised oscillator strength (GOS) is sensitive to the choice of the model atomic potential. In electron-impact ionisation, 'delayed maxima' of the GOS often occur near the ionisation threshold due to centrifugal barrier effects. In the present work, the sensitivity of this important effect of the centrifugal barrier to the model atomic potential has been studied using two local density atomic potentials, namely, the nonrelativistic Hartree-Slater potential and the relativistic RCB potential. Calculations have been done for electron-impact 2p ionisation of atomic oxygen, neon, aluminium, argon, iron and zinc. The GOS results for 2p ionisation of oxygen and aluminium are presented to illustrate the results.

Electron-impact collisions excite optically allowed as well as optically forbidden transitions. In this paper we report details of optical oscillator strengths calculated using the RCS potential. The oscillator strengths have been calculated using the small-momentum-transfer limit (K -+ 0) of the electron-impact generalised oscillator strength. Results for the Cooperminimum trajectories for some high-Z (82-90) atoms and for Kr 3p ionisation are reported. These have been found to be in fair agreement with results of previous calculations in which more involved techniques were used. The present results offer further support to the utility of the RCS potential in atomic physics by providing first estimates of some important atomic properties, using a relativistic atomic potential which is computationally not very demanding