Introduction to Quantum Mechanics for Materials Science
Quantum mechanics is an essential theoretical framework for understanding the electronic structure, chemical bonding, optical properties, and magnetism of materials. In this series, from the fundamentals of wave functions and Schrödinger equations to harmonic oscillators, hydrogen atoms, angular momentum theory, and perturbation theory, you will learn theory and numerical computation (Python/NumPy/SciPy) in pairs. The goal is to understand quantum effects in materials science and provide an introduction to first-principles calculations.
Basic knowledge of calculus and vector analysis (partial differential equations), linear algebra (eigenvalue problems, matrices) is required. Understanding the basics of Python is desirable.
Learn the fundamental principles of quantum mechanics, the interpretation of wave functions, stationary states, and exact solutions for one-dimensional potential well problems. Understand the Born interpretation, normalization, and probability density concepts, and implement methods to numerically solve eigenvalue problems with NumPy.
Learn the analytical solutions of quantum harmonic oscillators, Hermite polynomials, and creation/annihilation operators. Implement the principles of tunneling phenomena, calculation of transmission coefficients, and applications to the operating principles of scanning tunneling microscopy (STM).
Learn the hydrogen atom as a central force problem, the derivation of radial equations and Laguerre polynomials, and spherical harmonics. Implement the shapes of atomic orbitals (s, p, d orbitals), the physical meaning of quantum numbers, and visualization of electron clouds.
Learn the commutation relations of angular momentum operators, ladder operators, and quantum theory of spin angular momentum. Understand Pauli matrices and spin-1/2 systems, spin-orbit interactions, applications to magnetic materials, and implement angular momentum coupling.
Learn time-independent perturbation theory (non-degenerate and degenerate systems), variational principles, and the Rayleigh-Ritz method. Implement energy calculations for ground and excited states, applications to molecular orbital calculations, and an introduction to the Hartree-Fock method.
Upon completing this series, you will achieve:
For more advanced study in this field:
Expand your knowledge with related topics:
Apply your skills to hands-on projects: