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Quantum EngineeringYear 1: Quantum Mechanics CoreMonth 14Day 376

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Year 1·Month 14·Week 2

Day 376: Finite Square Well - Wave Functions and Penetration

Day 376 of 2,016~19 min read

Learning Objectives

  • •Construct explicit wave functions for even and odd parity bound states
  • •Apply normalization conditions to determine wave function amplitudes
  • •Calculate and interpret the penetration depth
  • •Compute the probability of finding the particle in the classically forbidden region
  • •Explain the relationship between binding energy and penetration
  • •Compare finite well eigenfunctions to their infinite well counterparts

Today's Schedule (7 hours)

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On this page

1 Complete Wave Function ConstructionEven Parity StatesOdd Parity States2 Matching Conditions and Amplitude RatiosEven Parity Continuity at x aOdd Parity Continuity at x a3 NormalizationEven Parity Normalization4 Penetration into Classically Forbidden RegionsThe Classically Forbidden RegionPenetration Depth5 Probability in Forbidden RegionFor Even Parity6 Energy Dependence of Penetration7 Wave Function Visualization8 Comparison Finite vs Infinite Well9 Effective Width and Lowered Energy10 Asymptotic BehaviorNear the Threshold E to 0-Deep Binding E to V_0Physical InterpretationQuantum Tunneling PrecursorThe Uncertainty Principle PerspectiveConnection to Alpha DecayQuantum Computing ConnectionQuantum Dot Wave FunctionsTunnel Coupling in Double DotsTransmon Penetration
Day 375Day 376 of 2,016Day 377