Problem 1. Consider a square well that extends from 0 to L.
(a) Write down the general solution for the wave function inside the well.
(b) Determine the specific solutions inside the well for the ground state and for the rst excited state by applying the boundary conditions at x = 0 and at x = L.
Now consider a 50:50 superposition of the ground state and the rst excited state.
(c) Write down the normalized wave function for this superposition state.
(d) Explain the time-dependence of the ground state wavefunction in the complex plane.
(e) Explain the time-dependence of the rst excited state wavefunction in the complex plane.
(f) Explain the time-dependence of the 50:50 superposition state wavefunction in the complex plane.
(g) Explain the time-dependence of the 50:50 superposition state probability density in the complex
plane.
(h) What is the oscillation frequency of the 50:50 superposition state probability density?

This solutions contains step-by-step calculations to determine the wave function and the time dependence inside the well. It also calculates the specific solutions for the ground states and excited state and takes into account a 50:50 superposition situation.

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