Problem 12.2

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A particle is in a one-dimensional harmonic oscillator potential (ħ = 2m = ω = 1; 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 E_n?  How does this agree with or disagree with the standard case for the harmonic oscillator? 

4. How much kinetic and potential energies are in an arbitrary energy state?

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