This book is aimed squarely at B.Sc. or B.A. undergraduate students of mathematical physics, experimental physics or theoretical chemistry. The book is designed to lead the reader through the famous problems that have made quantum mechanics the accurate description of our world that it is. Physical problems in Quantum Mechanics is resolutely mathematical in nature, as it should be, since only through a mathematical description of these phenomena can they be understood. The route that would have been taken in the 1920s and 1930s is followed, in that the differential equations of Schrödinger are solved, each problem being defined by the appropriate potential energy function and boundary conditions. The solutions thereby are examined and the consequent properties of the system at hand (Energy, angular momentum, transition probabilities etc.) are determined. Systems examined include the particle in one, two and three dimensional spherical boxes, the hydrogen atom in three dimensions and the harmonic oscillator. The book examines the solutions in both the position and momentum bases, drawing conclusions as to the meaning of those solutions in both bases. Bonding in the Hydrogen molecule is also examined in an exact mathematical fashion, following the route taken by the pioneers in Göttingen in the early 1920s by solving the two electron Schrödinger Wave Equation. Therein, electron-proton attraction and electron-electron repulsion integrals are computed exactly to obtain the molecular energies and their corresponding wavefunction solutions. Finally, the Dirac equation is examined to directly reveal the nature of the electron, the predicted partner particle, the positron, and the prediction of the existence of spin on both the electron and the positron, a momentous moment in the history of 20th century science.