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Quantum Atomic Structure — Wave Functions, Orbitals & Spin Explained | Chapter 41 of University Physics Chapter 41 extends quantum mechanics into three dimensions, applying Schrödinger’s equation to atomic systems. Explore how quantum numbers define wave functions and orbitals, and uncover the roles of spin, magnetic fields, and entanglement in shaping atomic behavior. Be sure to watch the full video summary for animations of atomic orbitals and spectral line splitting. 3D Schrödinger Equation & Atomic Wave Functions In three dimensions, the time-independent Schrödinger equation −(ħ²/2m)∇²ψ + U(r)ψ = Eψ governs atomic wave functions ψ(x,y,z). These stationary states, when normalized, satisfy ∫|ψ|² dV = 1, giving the probability density for locating electrons in space. Particle in a Cubical Box A particle confined to a cube of side L has energy levels: E = (ħ²π²/2mL²)(n x ² + n y ² + n z ²) , with quantum numbers (n x ,n y ,n z ) = 1,2,3… Degenerate states s...

Atomic Structure, Interatomic Bonding, and Crystalline Solids — Chapter 2 Summary from Callister & Rethwisch Delve into the fundamentals of how materials behave at the atomic level with this summary of Chapter 2, “Atomic Structure, Interatomic Bonding, and Crystalline Solids,” from Materials Science and Engineering: An Introduction by William D. Callister, Jr. and David G. Rethwisch. This resource breaks down the essential principles of atomic structure, the types of chemical bonds that determine material properties, and the organization of atoms in solid materials. Whether you’re studying for an exam or building a foundation in materials science, this guide is designed for clarity and depth. Want a clear audio summary? Watch the podcast episode below, and subscribe to Last Minute Lecture for comprehensive chapter-based study guides! Understanding Atomic Structure At the heart of materials science is the atom. Each atom consists of a nucleus—containing protons and ...