Quantum Harmonic Oscillator: Power Series and Ladder Operators

The quantum harmonic oscillator, worked out from the Schrödinger equation to coherent states, with every step animated! We start with why almost any smooth potential well behaves like a spring near its minimum, then solve the oscillator two ways. The power-series method shows why only the energies E = ℏω(n + ½) are allowed. They are evenly spaced, and even the lowest one, ½ℏω, is above zero. Ladder operators then reach the same energy levels and the same states with algebra alone. After that, the video looks at the ground state's minimum uncertainty, why the density of an energy eigenstate never changes, how a superposition of two levels moves back and forth, and why a coherent state's center follows the classical orbit while its width stays fixed. It ends with normal modes, which break small vibrations of molecules, crystals and fields into independent oscillators of this same kind. Further reading (free) David Skinner, Principles of Quantum Mechanics, ch. 3, The Harmonic Oscillator (Cambridge): https://www.damtp.cam.ac.uk/user/dbs2... Barton Zwiebach, MIT 8.04 Quantum Physics I (MIT OpenCourseWare): https://ocw.mit.edu/courses/8-04-quan... Richard Fitzpatrick, Quantum Mechanics, Simple Harmonic Oscillator: https://farside.ph.utexas.edu/teachin... James Freericks, Operator Solution of the Harmonic Oscillator: https://site.physics.georgetown.edu/~... Michael Fowler, Coherent States of the Simple Harmonic Oscillator: https://galileo.phys.virginia.edu/cla... David Tong, Lectures on Quantum Field Theory, section 2, Free Fields: https://www.damtp.cam.ac.uk/user/tong... The animations are drawn in Python directly from the equations shown. More from Blake Pi Website: https://blakepierpoint.com/ Instagram:   / blakepiphysics   Substack: https://substack.com/@blakepierpoint #QuantumMechanics #Physics #HarmonicOscillator