## ABSTRACT

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.