Quantum Mechanics - Harmonic Oscillator - DERIVATION (Analytic Method) | Energy & Eigenfunction Solutions

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for the love of physics physics education lectures quantum mechanics quantum harmonic oscillator schrodinger equation hermite polynomials gaussian wavefunction quantum mechanics derivation harmonic oscillator harmonic oscillator derivation

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Step by Step Derivation of Energy Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator from the Schrodinger Equation (using Analytic Method) - Final Solutions are Hermite Polynomials modulated by Gaussian Function 𓏬𓏬𓏬𓏬𓏬QM Lecture Series Playlist𓏬𓏬𓏬𓏬𓏬 https://youtube.com/playlist?list=PLRN3HroZGu2mCtdalEmZAM2nr1xBWAtUn&si=8ejaL8RrSFQGbtzx 𓏬𓏬𓏬𓏬𓏬ELEVATE CLASSES𓏬𓏬𓏬𓏬𓏬 Elevate your preparation with Elevate Classes. Find LIVE & Recorded Courses for Physics IITJAM, CSIR-NET, GATE, TIFR, JEST etc on our platform. Use Coupon LOVEPHYSICS to get a 5% OFF in any course. Website ► https://www.elevateclasses.in/ Android ► https://bit.ly/3zU71ur iOS ► https://apple.co/3ZPRWVJ (use ORG code - AHGXS) 𓏬𓏬𓏬𓏬𓏬LEC NOTES - GDRIVE𓏬𓏬𓏬𓏬𓏬 Find the PDF Scanned copy of my NOTES for this lecture here: https://drive.google.com/file/d/11StclzVKxaimWx1tfIeICGUC4HQcoO1w/view?usp=sharing 𓏬𓏬𓏬𓏬𓏬VIDEO DESCRIPTION𓏬𓏬𓏬𓏬𓏬𓏬 In this video, I mathematically derive the Energy Eigenvalues and the detailed Eigenfunction solutions of the Harmonic Oscillator using Schrodinger's Equation. I use the Analytic Method, that involves Power Series solutions that lead to Hermite Polynomials, which are ultimately modulated by a Gaussian function. Timestamps of the Full Derivation as follows 00:00 Introduction 05:42 Schrodinger Equation - Variable Substitution 12:42 Asymptotic Behavior - Gaussian Function for large x 18:46 Trial Function for small x 25:35 Power Series Solution 33:20 Recursion Relation & Asymptotic Behavior 46:48 Truncation Condition & Energy Eigenvalues 58:49 Hermite Polynomials 1:19:15 Final Results & Plots 𓏬𓏬𓏬𓏬𓏬𓏬TELEGRAM𓏬𓏬𓏬𓏬𓏬𓏬 Join my Telegram Channel ► https://t.me/FortheLoveofPhysicsYT 𓏬𓏬𓏬𓏬𓏬SUPPORT CHANNEL𓏬𓏬𓏬𓏬𓏬 Your Financial support provides me an additional incentive to create high quality lecture videos. I am very much thankful for your generosity and kindness Support/ Patreon ❤️❤️❤️https://www.patreon.com/dibyajyotidas Donate/ Paypal 🔥🔥🔥 https://paypal.me/FortheLoveofPhysics JOIN MEMBERSHIP in Youtube 😇😇😇 https://www.youtube.com/channel/UCOfLm6gZGt3vwTMKRg-irhg/join

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