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Problem 12.2: A particle is in a 1-d dimensionless harmonic oscillator potential

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A particle is in a one-dimensional harmonic oscillator potential (ħ = 2m = 1; ω = k = 2). The states shown are normalized. Shown are ψ and the results of the integrals that give <x> and <x2> and <p> and <p2>.  Vary n from 1 to 10.

  1. What do you notice about how <x> and <x2> and <p> and <p2> change?
  2. Calculate ΔxΔp for n = 0. What do you notice considering ħ = 1?
  3. What is En?  How does this agree with or disagree with the standard case for the harmonic oscillator?
  4. How much average kinetic and potential energies are in an arbitrary energy state?
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