Chapter 7: Quantum Theory

The theory emerges from empirical observations that energy is transferred in discrete packets and that matter exhibits dual particle-wave characteristics, formalized through the de Broglie relation connecting wavelength to momentum. At the heart of quantum mechanics lies the wavefunction, a mathematical function that contains all dynamical information about a system, with the Born interpretation providing the crucial link between this abstract mathematical object and physical reality by defining the probability density of finding a particle in a given region of space. Observable properties of quantum systems are extracted through operators, establishing the fundamental correspondence between mathematical operators and measurable physical quantities. The chapter then applies these principles to three essential types of motion encountered throughout chemistry and physics. For translational motion, the particle in a box model reveals how quantization naturally emerges from applying boundary conditions to the Schrödinger equation, introduces the concept of zero-point energy as a non-classical feature, and demonstrates how multi-dimensional systems can be constructed from products of one-dimensional solutions while generating degeneracy when multiple quantum states share identical energy levels. Quantum tunneling is presented as a non-classical phenomenon where particles penetrate regions forbidden by classical mechanics, a process critical to many chemical and nuclear processes. Vibrational motion is modeled using the quantum harmonic oscillator, which confirms the existence of evenly spaced quantized energy levels and demonstrates that molecular vibrations never truly cease, even in the ground state. Finally, rotational motion of particles confined to a ring or spherical surface introduces angular momentum as a quantized property and establishes the quantum numbers governing rotational states, providing essential concepts for understanding molecular rotation and atomic structure.