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...

Quantum Nature of Matter – Electron Waves, Bohr Model & Light Explained | Chapter 39 of University Physics Chapter 39 delves into how matter—especially electrons—exhibits both wave and particle properties, laying the groundwork for quantum mechanics. You’ll learn about de Broglie wavelengths, atomic spectra, the Bohr model, laser operation, blackbody radiation, and the Heisenberg uncertainty principle. Watch the full video summary here to see animations of electron diffraction and quantum effects. Electron Waves & de Broglie Hypothesis Louis de Broglie proposed that any particle with momentum p has a wavelength: λ = h / p . Electron diffraction experiments confirm this wave nature—electrons accelerated through a potential V have: λ = h / √(2 m e V) , enabling electron microscopes (TEM, SEM) to achieve atomic-scale resolution. Atomic Structure & the Bohr Model Classical physics failed to explain why atoms are stable and emit discrete spectral lines. Bohr int...

Photon Model & Quantum Phenomena Explained | Chapter 38 of University Physics Chapter 38 unveils light’s particle nature, introducing photons and key quantum effects that classical physics cannot explain. You’ll learn how photons eject electrons in the photoelectric effect, scatter off particles in Compton collisions, create matter–antimatter pairs, and obey fundamental uncertainty limits. Watch the full video summary for visual demonstrations and deeper insights. Photoelectric Effect & the Photon Model Classical waves couldn’t explain why below a threshold frequency no electrons are emitted. Einstein proposed light travels in quanta— photons —with energy E = hf = hc/λ . When a photon of energy hf exceeds a material’s work function φ , it ejects an electron with maximum kinetic energy: K max = hf – φ = eV₀ Light intensity controls the number of photons (and thus electrons), not their individual energy. X-Ray Production & Bremsstrahlung In X-ray tubes, ...