Fundamentals of Quantum Mechanics (Dirac-Formalism)

Ph. Droz-Vincent

S. J. L. van Eijndhoven, J. de Graaf

Manuel Gadella, F. Gómez-Cubillo

Manan Sengupta

The concept of constrained Hamiltonian dynamical systems and their quantization, introduced by Dirac, constitutes one of the most fundamental contributions to the development of the canonical formalism in recent times. Dirac’s concepts were later incorporated into the path-integral formulation of quantum mechanics by Faddeev, which played an important role in understanding some subtle aspects in the quantization of non-Abelian gauge theory. An example is given from nonrelativistic quantum mechanics which illustrates the formalism through simple analogy.

R. Jagannathan, Sameen Ahmed Khan

This chapter reviews quantum mechanics required to pursue the quantum charged particle beam optics, the central theme of this book. All physical phenomena are quantum mechanical at the fundamental level. Classical mechanics observed in the macroscopic world is an approximation. That is why it raises a curiosity when classical mechanics is found to be very successful at a microscopic level in the design and operation of beam devices like an electron microscope or a particle accelerator. This chapter gives an introductory review of nonrelativistic and relativistic quantum mechanics just at the level necessary for presenting the formalism of quantum charged particle beam optics. All the basic equations of quantum mechanics, Schrödinger’s time-evolution equation, the Klein-Gordon equation, and the Dirac equation, are given in detail. The Feshbach-Villars formalism of the Klein-Gordon equation and the Foldy-Wouthuysen transformation of the Dirac equation, forming the basis for the development of quantum charged particle beam optics, are described.

Juan José García‐Ripoll

In this chapter, we review basic concepts from quantum mechanics that will be required for the study of superconducting quantum circuits. We review the fundamental idea of energy quantization and how this can be formalized, using Dirac's ideas, to develop a quantum mechanical description that is consistent with the classical theory for a comparable object. We review the notions of quantum state, observable and projective and generalized measurements, particularizing some of these ideas to the simple case of a two-dimensional object or qubit.