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Quantum Engineering Year 1: Quantum Mechanics Core Month 14Day 368

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Year 1·Month 14·Week 1

Day 368: Gaussian Wave Packets

Day 368 of 2,016~15 min read

Learning Objectives

•**Write the normalized Gaussian wave packet** in position and momentum space
•**Prove the minimum uncertainty relation** Δx·Δp = ℏ/2 for Gaussians
•**Compute all expectation values** ⟨x⟩, ⟨p⟩, ⟨x²⟩, ⟨p²⟩ analytically
•**Explain why Gaussians are special** from multiple perspectives
•**Connect Gaussians to coherent states** in quantum optics
•**Construct Gaussian wave packets** with specified properties

Today's Schedule (7 hours)

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1 The Standard Gaussian Wave Packet 2 General Gaussian Wave Packet 3 Fourier Transform Gaussian to Gaussian 4 Expectation Values 5 Uncertainties 6 Minimum Uncertainty 7 Why Gaussians Are Special Mathematical Properties Physical Properties 8 Coherent States Connection 9 Probability Distributions 10 Wigner Function Quantum Computing Connection Gaussian States in Continuous-Variable QC Quantum Error Correction

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