Quantum Mechanics: The Self-assembly of Atoms

Quantum mechanics is the physical basis for the self-assembly of the Universe. We base the development of quantum mechanics on a set of axioms, beginning with complex linear vector spaces (Hilbert space) and the symmetries of space–time as transformations that preserve probability in terms of the length of state vectors. We show that confinement of objects at the quantum scale results in the emergence of discrete energy states. Simple solutions of Schrödinger’s equation include the free particle, the particle in a box, the harmonic oscillator, and the hydrogen atom. The uncertainty principle is derived, along with selection rules, and zero-point motion. Another source of energy for enzyme catalysis is the borrowing of energy from the environment allowed by the uncertainty principle. Commutators, symmetry, and “group theory” are introduced. The “harmonic oscillator” is revisited in terms of creation and annihilation operators, which are important in their own right and also as a tool to understand the interaction of light with matter. Rotational symmetry and angular momentum are shown to be intimately related.