Harmonic Oscillator – DERIVATION (Analytic Method) | Energy & Eigenfunction Solutions
22.5K views on YouTube
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
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