Quantum Mechanics: Potential Well

A particle is trapped in a one-dimensional region of space by a potential energy function which is zero between positions zero and L, and equal to U₀ at all other positions. This is referred to as a potential well of depth U₀.
Examine a proton in a potential well of depth 50 MeV and width 10 x 10^-15 m.

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Question 1: Infinite Well
If the potential well was infinitely deep, determine the ground state energy. Is this also the ground state energy in the finite well?

E_1(infinite well) = (1)^2(h)^2/(8*(mass of proton)*(10*10^-15) = 2.05MeV
E_1(finite well) = 1.8MeV

The ground state energy of an infinite well is more than the ground state energy of a finite well.

Question 2: First Excited State
If the potential well was infinitely deep, determine the energy of the first excited state (n = 2). Is this also the energy of the first excited state in the finite well?

E_2(infinite) = 4E_1 = 8.20MeV.
E_2(finite) = 6.8 MeV
The energy of the first excited state in the finite well is not the same as the one in the infinite well.

Question 3: "Forbidden" Regions
Since the wavefunction can penetrate into the "forbidden" regions, will the energy of the first excited state in the finite well to be greater than or less than the energy of the first excited state in the infinite well? Why?

The energy in the first excited state in the finite well is less than the first excited state in the infinite well due to the greater probability of tunneling.

Question 4: More Shallow Well
Will the energy of the n = 3 state increase or decrease if the depth of the potential well is decreased from 50 MeV to 25 MeV? Why?

The energy of the n=3 state decreases if the potential well is decreased from 50MeV to 25MeV due to less tunneling.

Question 5: Penetration Depth
What will happen to the penetration depth as the mass of the particle is increased?
As the mass of the particle increases the penetration depth decreases