Final preparation problems.

Please keep posting more suggestions for problems. If you collaborate and participate then it is more likely that the final will not be incomprehensible and problems that are difficult to understand or pitched at the wrong level.

I'll post some easier problems later. I am still waiting for more feedback in the post "What would you like to see on the final."

1) Consider a graphene sheet that is 2 mm long and 4 nm wide. Suppose that you dope the right half (Let's call that x>0) with donors to get a carrier density of 10^12 carriers/cm^2. Now suppose you role that into a 2mm long, narrow tube.
a) what is the radius of the tube?
b) what is the linear carrier density
c) If you apply an electric field that gives an (average) speed of 10^7 cm/sec to the carriers moving to the left, how many will arrive at the x=0 boundary per second?

2) With regard to the same half-doped graphene tube:
a) can the moving electron coming from the doped x>0 side cross the x=0 boundary? Why or why not? (the left half, x < 0, is undoped.)
b) will they fall into the valence band? why or why not?
c) If not, what do you need to do to enable them to drop into the valence band?
d) assuming you have solved part c) and that each one dropping into the valence band emits a photon,
how much radiated power would you get from one such "nanotube"... if the band gap were engineered to be 0.5 eV.

3. Previously we discussed that in the presence of an oscillating E-field of just the right frequency, the probability of a transition from one quantum state to another is proportional to the electric field squared times an integral of the wave-function of the initial state, the wave-function of the final state and x (where the x factor comes from the interaction potential $e E_x x$ (for an electric field polarized in the x direction).  Starting from this, discuss how a laser works. Compare and contrast a laser and and LED made with the same semiconductor. What are the similarities and differences?
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4. extra credit: a) Write the equation for a crystal state of wave-vector k (assuming you know the atom state that the crystal state evolves from). For a bandwidth of 4 eV, what is the velocity associated with theis crystal state? (Do you need to be given anything else to calculate that?)
c)  Discuss what the animation below shows. [Hint: it is a Gaussian wave packet of crystal states center NOT at k=0. The packet is pretty narrow. This is for a 1D crystal with b=0.1 nm. Discuss: Around what k-value is this packet centered. How wide is it (in k and in x)? and other relevant things. Explain your reasoning. Please feel free and encouraged to discuss this here (in the comments).



 5. Consider an electron in a harmonic oscillator potential. Suppose that at t=0 it is in the state
$\Psi (x,0) = (c_o + c_1 x + c_2 x^2) e^{-x^2/2a^2}$.
a) Discuss the time dependence of this state? Is it time dependent?
b) How would you calculate $\bar{x}$? Would it depend on time?

6. Referring to hydrogen atom states,
a) write the state $\psi_{2,1,3/2,1/2}$ in terms of $\psi_{n.l.m}$ states (with spin).
b) extra credit: write the state $\psi_{2,1,3/2,1/2}$ in terms of $\psi_{2x}$ type states (with spin).

7. Suppose an electron in a hydrogen atom is in a state that is an equal mix of the ground state and 1st excited state. Pick a particular 1st excited state and for your choice:
a) Describe/discuss what you would expect to get if you calculated $\bar{x}$,  $\bar{y}$ and $\bar{z}$ (include a graph or graphs) in your description.
b) Calculate $\bar{x}$,  $\bar{y}$ and $\bar{z}$ (include a graph or graphs).

8. Describe a p-n junction and how it can be used to create an LED or laser. Which way do the electrons flow? what happens? etc  extra credit for elucidating the difference between an LED and a Laser.

9. For an electron in a harmonic oscillator potential that is in a state that is an equal mix of the ground state and 1st excited state,
a) calculate and graph $\bar{x}$? Would it depend on time?

10. Same as 9, but instead of an equal mix suppose that the coefficient of the g.s. term is 0.9.
a) calculate and graph $\bar{x}$? Compare your results from 9 and 10.

10.1  Same as 9, but instead of an equal mix suppose that the coefficient of the g.s. term is 0.99.
a) What is the coefficient of the 1st-excited state?
b) calculate and graph $\bar{x}$? Compare your results from 9 and 10.

11. Suppose you have a biased (forward) p-n junction with a junction area of 0.1 cm^2. If the average electron speed is 5 x 10^6 cm/sec and the electron density is 10^17/cm^3, how much power would be radiated in either an LED or Laser (assume 100% efficiency). You can assume an equal and opposite flow of holes from the p side to the n side.

12. For an electron in the ground state of an infinite square well of width 0.5 nm, what is the expectation value in eV of:
a) the kinetic energy,
b) the potential energy.
c) why does an electron in a well have kinetic energy?

13. Consider a finite square well of depth 20 eV that has exactly (only) 3 bound states. One is 2.5 eV above the energy of the bottom of the well, the next 9 eV (same reference) and the 3rd at 17 eV.
a) sketch the states.
b) Describe and sketch the optical absorption spectrum that you would predict for this system. Label all sharp lines corresponding to transitions with their initial and final states and indicate their energy.
c) Assuming the strength of any transition is related to x integrated with the initial and final states, which transitions would be unobservable?