Ever become frustrated when the textbook skip and jump steps in formulas derivation in the teaching of important assumption and approximation? If you are like countless students being stuck at, a particular mathematical calculus-working step or steps while reading a simple undergraduate physics text, then this manuscript may help in your journey. For the completeness to accompany the undergraduate introduction text of Liboff’s on Introductory Quantum Mechanics, 4th Edition study, that includes calculus of partial differential equations, quantum mechanics Dirac bra-ket equation, and integration by parts. This text provides a detail step-by-step symbolical derivation workouts that are omitted or incomplete in between the presented physics formulas or definitions. All the mathematical derivations and expansions involve the trigonometry functions and identities, first and second order time independent and time dependent partial differential equations, calculus of variations as well as simultaneous equations solving via matrix or direct substitution method. Readers with/without fundamental handle on quantum mechanics and/or undergraduate level mathematics proficiency but wish to study the physics and/or the applied mathematics, can now use this text as a step-by-step systematical fill-in-the-blank reference and derivation counter-checking resources. Readers can follow and learn the essence of quantum mechanics. There are 8 chapters in the part 1 of 2 series of undergraduate quantum mechanics elementary principles and applications 1D problems to be examined and derived accordingly as discussed inside the Liboff’s Introductory Quantum Mechanics, 4th edition text. They are namely the Review of Concepts of Classical Mechanics, Historical Review on experiments and theories, The Postulates of Quantum Mechanics operators, eigenwaves and eigenenergies, Preparatory Concepts in function spaces and Hermitian operators, Superposition and Compatible Observables, Time Development, Conservation Theorems and Parity, Additional 1D Problems of bound and unbound states, and Finite Potential Well, Periodic Lattice and sample problems of 2 degrees of freedom.