Quantum Physics and Modern Applications

The following sections are included:

• Preface
• Contents

This chapter contains primarily problems in basic quantum mechanics. It covers the numerous aspects of quantum mechanics as follows: Schrodinger equations, eigenfunctions and eigenvalues, completeness, and so on. The concepts of non-commutativity and Hermiticity in physics are also introduced. There are problems of harmonic oscillations which also introduce the matrix formulation of quantum mechanics. In the Dirac formalism, discussion hovers around its representation in general, matrix and function spaces. This chapter is suitable as a supplementary material for students learning quantum mechanics at the level of undergraduate. However, problems and discussions around the Dirac formalism may also be useful for deeper thought and advanced understanding.

Chapter 2 is dedicated to spinor formalism in quantum mechanics. Spin is presented as a separate chapter because spinor formalism is widely applied in emerging fields like spintronics and graphene, as well as topological systems like topological insulators and Weyl and Dirac systems. It is also a crucial tool for the study of spin-based Berry curvature and the non-abelian quantum transport in the formulation of non-equilibrium Green’s function. On top of that, the field of quantum computation and information too is primarily run on spinor physics. These are all new research fields that hover heavily around spinor physics.

Second quantization is the foundation of quantum field theory. Recently, it has also been used very extensively in research fields like quantum transport, quantum optics, quantum nanoscience, and so forth. This chapter deals with second quantization and its applications in mainly the context of nanoscience and condensed matter physics. Particular attention is also paid to its application in the non-equilibrium Green’s function — a modern topic for quantum transport in electronic and spintronic devices. It is suitable for postgraduate or senior undergraduate students taking courses in advanced quantum mechanics.

This chapter deals with the non-equilibrium Green’s Function (NEGF) — a modern topic with problems designed to suit applications in nanoscale electronic and spintronic devices. As Green’s functions in quantum mechanical context are built on second quantization, requisite knowledge of second quantization is needed and Chapter 3 would come in handy. In this chapter, Green’s functions are first introduced in the context of general applications in condensed matter systems. It is followed by a full-fledged non-equilibrium formulation for the study of quantum transport in electronic as well as spinor-based devices. The latter part of this chapter is particularly specialized and problems are formulated for research purposes. In general, problems here could be used to augment the numerous courses of postgraduate quantum mechanics.

This chapter is about gauge and topology with modern emphasis on applications in new materials like graphene and topological systems. Coordinate transformation is introduced to familiarize readers with the frame of references and their physical consequences. Metric of space is implied but not explicitly dealt with. The Berry–Pancharatnam phase is introduced in applications like graphene and semiconductor systems. Towards the end, the topological nature of phase and curvature are introduced with special attention paid to numerous models in condensed matter systems, e.g., the Su–Schrieffer–Heeger (SSH) model and polarization in the 2D Chern insulator. This is a specialized topic suitable for research purposes. Problems here could also be used for postgraduate quantum mechanics.

This chapter comprises numerous sub-topics in condensed matter physics. It consists of conventional topics like the band structure, as well as selective problems of heat capacity and magnetic susceptibility. Towards the end, the modern and advanced topics of entanglement entropy are presented in numerous different contexts. These are advanced sub-topics and problems in this chapter could also be used for postgraduate courses in quantum mechanics and condensed matter physics.

This is a cross-disciplinary chapter that covers miscellaneous topics from general to mathematical physics. Problems here are not necessarily pure quantum by nature, but many deal with issues that have inspired the development of quantum mechanics and quantum fields. For example, the concept of minimization leads to the Euler–Lagrange equations that also form the foundation of quantum field theory. Questions are also raised in problems to stimulate thought about physical issues based on quantum physics, e.g., visualization of the electrons.

Chapter 8 is dedicated to the field of Quantum Computation and Information (QCI). The required quantum mechanical foundation of spin could be found in Chapter 2. This chapter begins with the introduction of the density matrix and its statistical and entanglement connotation. On the application side, the quantum gates of NOT, Hadamard, and CNOT are introduced. The concepts of quantum parallelism, entanglement, and no-cloning are introduced prior to the design of quantum circuits. The latter part of this chapter would introduce more advanced circuit designs like the teleportation and the implementation of algorithm. On the whole, this chapter can be used to teach quantum mechanics while showcasing its applications in QCI at the same time.