Last Minute Lecture

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...

Wave Functions, Tunneling & Schrödinger Equation Explained | Chapter 40 of University Physics Chapter 40 dives into the core of quantum mechanics: wave functions, Schrödinger’s equation, particle confinement in potential wells, quantum tunneling, and the quantum harmonic oscillator. Whether you're studying atomic systems or advanced quantum devices, this breakdown will clarify these fundamental concepts. Watch the full video summary on YouTube to see animated solutions of the Schrödinger equation and tunneling demonstrations. Wave Functions & Schrödinger’s Equation The wave function Ψ(x,t) encodes all information about a quantum system, with probability density given by |Ψ|². Its evolution follows the time-dependent Schrödinger equation: −(ħ²/2m) ∂²Ψ/∂x² + U(x)Ψ = iħ ∂Ψ/∂t . Superpositions of eigenstates form wave packets that describe localized particles, but they spread over time due to the uncertainty principle (Δx·Δp ≥ ħ/2). Particle in a Box (Infinite S...