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1) Consider aparticle of mass m confined by the following potential:
a) What is the wavefunction for |x| > L/2?
b) Consider the time independent Schrodinger equation:
What quantity from classical mechanics do the first and second quantity in the left side of the equation most closely represent?
c) First, independent of the boundary conditions, what is the most general solution that solves the Schrodinger equation above in the region where the potential vanishes.
d) State the boundary conditions that the wave function must satisfy.
e) Derive and normalize expressions for the allowed energies and energy eigenfunctions.
f) Sketch the first three eigenfunctions.
2) Consider the following one-dimensional potential function:
Assume the total energy of an electron is E < V0 .
a) Write the wave functions that apply in each region.
b) State the boundary conditions that the wave function must satisfy.
c) Show explicitly why, or why not, the energy levels of the electron are quantized.
3) List and explain at least three phenomena that can’t be explained by classical physics but can be explained by quantum physics. You can answer in this form. “classical physics predicts that … ., but the contradiction is …., quantum physics solves this problem by … .”