Tunneling Time and Probability of an Electron in a Square Well

Imagine an electron trapped in a well that has walls that are 5nm thick and 10eV high and is 10nm wide. calculate the probability of tunneling for an electron with 0.5eV of kinetic energy. Now, determine the number of times the electron would hit a wall each second. using these two pieces of information determine how long it would take for there to be a 50% probability that the electron has already tunneled out. (imagine this as if you had a large number of these wells and you observed that at time, T, 50% of them were empty.

Solution Preview

After each collision with a wall, the electron has a probability of e^(-kappa T) of tunneling out, where
T = 5 nm is the thickness of the wall,

kappa = sqrt(2m(V0-E))/hbar

is the tunneling coefficient, m = 9.11 * 10^-31 kg is the mass of the electron, V0 = 10 eV is the height of the walls, and E = 0.5 eV is the kinetic energy of the electron. Plugging in the numbers in SI units, we have

We calculate the tunneling probability of an electron of given kinetic energy in a square well of given dimensions as well as the expected time for the electron to escape from the well.

Given a tunneling barrier with a thickness of 2nm and a barrier height of 5eV, what is the minimum kinetic energy an electron would have to have to have a 50% chance of passing through? (hint it doesn't necessarily have to be less than the barrier height).
I need the step-by-step solution please.
Thank You.

Please help me stepwise in word / pdf.
Given a tunneling barrier with a thickness of 2nm and a barrier height of 5eV, what is the minimum kinetic energy an electron would have to have to have a 50% chance of passing through? (hint it doesn't necessarily have to be less than the barrier height)

An electron is accelerated through potential difference of 3eV and is incident on a finite potential barrier of height 5eV and thickness 5x 10^-10m. What is the probability that the electron will tunnel though the barrier.
I am not 100 percent sure how to do this so a detailed solution would be helpful.

A Hydrogen atom and a Helium atom each with 4 eV of KE approach a thin barrier 6 MeV high which has the greater probability of tunneling through the barrier? Explain.

Look up the magnetic dipole moment of a hydrogen atom and the bohr radius, and use the assumption of uniform circular motion to calculate the effective current with in a hydrogen atom. Compare this current to that of a scanning tunneling microscope 1nA. Which is greater and by how many orders of magnitude?
Magnetic dipole m

See the attached file.
1. What is meant by an expectation value?
2. What does it mean when we say that 'mod Psi-square' is a 'probability density', as opposed to a probability?
3. What is the difference between an operator that operates on an eigen function and one that operates on any function?
4. What is meant when

Please explain what proton tunneling is and how to compute the probability of tunnelling and the loss in energy as a result. Please give an example.
I would like a very clear and complete explanation.

What is the probability that an electron in the infinite well in the state Un(x) =[(2/L)^.5]*sin(Pi*n*x/L) is found in the region between x = 0 and x = L/2 , where the Un are the eigenfunctions of the infinite-well potential?

5. What is the quantum mechanical understanding of the electron bound to a nucleus in an atom?
What is an orbital?
What is the physical significance of an orbital?
Please see attached file for full problem description.