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Quantum Engineering Year 1: Quantum Mechanics Core Month 16Day 426

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Year 1·Month 16·Week 1

Day 426: 3D Isotropic Harmonic Oscillator

Day 426 of 2,016~10 min read

Learning Objectives

•Solve the 3D harmonic oscillator using separation in spherical coordinates
•Derive the energy spectrum E = ℏω(n + 3/2)
•Calculate degeneracies using combinatorics
•Understand the connection to Cartesian separation
•Recognize the hidden SU(3) symmetry
•Apply to molecular vibrations and trapped atoms

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The Potential Two Approaches Cartesian Solution Energy Spectrum Degeneracy Spherical Solution Radial Equation Radial Wavefunctions Relation Between Quantum Numbers Angular Momentum Content Verifying Degeneracy Hidden Symmetry SU3 CreationAnnihilation Operators Quantum Computing Connection Bosonic Modes Continuous Variable QC Trapped Ions

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