Quantum mechanics forms the foundation of all modern physics‎,‎ including atomic‎,‎ nuclear‎,‎ and molecular physics‎,‎ the physics of the elementary particles‎,‎ condensed matter physics‎.‎ Modern astrophysics also relies heavily on quantum mechanics‎.‎ Quantum theory is needed to understand the basis for new materials‎,‎ new devices‎,‎ the nature of light coming from stars‎,‎ the laws which govern the atomic nucleus‎,‎ and the physics of biological systems‎.‎ As a result the subject of this book is a required course for most physics graduate students‎.‎ While there are many books on the subject‎,‎ this book targets specifically graduate students and it is written with modern advances in various fields in mind‎.‎ Many examples treated in the various chapters as well as the emphasis of the presentation in the book are designed from the perspective of such problems‎.‎ For example‎,‎ the book begins by putting the Schrodinger equation on a spatial discrete lattice and the continuum limit is also discussed‎,‎ inspired by Hamiltonian lattice gauge theories‎.‎ The latter and advances