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The purpose of this problem is to compare the time dependencies for systems in a superposition...

The purpose of this problem is to compare the time dependencies for systems in a superposition of two energy eigenstates in an infinite square well to those in a simple harmonic oscillator.
Consider two systems (an infinite square well and a simple harmonic oscillator) that have the same value for their ground state energy Eground.

1) What is E3, the energy of the 2nd excited state (the third lowest energy) of the infinite square well system in terms of Eground? E3 =

2) What is E3, the energy of the 2nd excited state (the third lowest energy) of the simple harmonic oscillator system in terms of Eground? E3 =

3) Now suppose the wave function for the infinite square well system is a superposition of two energy eigenstates, namely its ground state and its second excited state. Assuming Eground = 20 eV, what is t1, the minimum time it takes for the probability density ρρ(x,t1) of the infinite square well system to return to its original value (ρρ(x,0)) at t = 0?

t1 =

4) Now suppose the wave function for the simple harmonic oscillator system is a superposition of two energy eigenstates, namely its ground state and its second excited state.
Assuming Eground = 20 eV, what is t1, the minimum time it takes for the probability density ρρ(x,t1) of the simple harmonic oscillator system to return to its original value (ρρ(x,0)) at t = 0?
t1 =

5) Now, both the infinite square well system and the simple harmonic oscillator system still have the same value for their ground state energies, Eground, but the wavefunctions for both systems are described as a superposition of two energy eigenstates, namely their ground state and their third excited state.
What is R(SHO/Square), the ratio of t1(SHO), the minimum time it takes for the probability density ρρ(x,t1,SHO) of the simple harmonic oscillator system to return to its original value (ρρ(x,0)) to t1(Square), the minimum time it takes for the probability density ρρ(x,t1,Square) of the infinite square well system to return to its original value (ρρ(x,0)) ? i.e., R(SHO/Square) = t1(SHO) / t1(Square).

R(SHO/Square) =

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