Homework 2, due friday Jan 17, 2 PM.

This is our last assignment on 1D QM and, as such, it is important to providing you with a foundation to understand the 3D quantum that we will do next. I would suggest starting soon and spending as much time as you can on this assignment.

 Whenever units are needed, please use eV and nm as your default units.  Problem 5 is pretty difficult, but I think that if you can do it and understand it then you will really learn a lot about the nature of potential and kinetic energy in quantum systems. This should be useful later in this class and, I think, in future quantum physics classes you take.

1. For an electron in the ground state of a 1D harmonic oscillator, calculate the potential energy expectation value.



2. For a harmonic oscillator system:
a) what are the units of k?
b) what value of k yields a ground state potential energy expectation value of 1 eV ? For this k value, what is the quantum length scale, "a"? (Recall that "a" is a parameter with units of length constructed from \( \hbar \), m and k.) (Use an electron mass for m.)
c) Calculate the expectation value of the kinetic energy and then evaluate it in eV for an electron in the ground state of this potential (i.e., a potential with this particular k value from part b)).
d) Calculate \( \Delta x\) in nm for an electron in the ground state of this potential. (Again, using the value of k from part b).)
e) (extra credit) What is \( \Delta x \Delta p\) for an electron in this state?

3.  For an infinite square well of width L (centered at x=0) calculate the kinetic energy of an electron
a) in the ground state
b) in the first excited state
c) which is larger?  by how much?

4. Calculate in eV the kinetic energy of an electron in the ground state of an infinite square well of width 0.613 nm.

5. For a finite square well centered at x=0, that is 0.613 wide (same as the previous problem) and 1.0 eV deep, I am pretty sure that the ground state is approximately as follows:
inside the well: $$\psi_1 (x) = (0.55)^{1/2} (2/L)^{1/2} cos(0.6 \pi x/L)$$
Outside the well on the right (that is for \(x \gt L/2 \) :
$$\psi_1(x) = (.193)^{1/2} (2/L)^{1/2} e^{-(x-L/2)/a_1}$$
where a = 0.27 nm
a) Sketch this state. 
b) Calculate the expectation value of the potential energy for an electron in this state. Where does the potential energy come from?
c) (extra credit) Calculate the expectation value of the kinetic energy for an electron in this state.
d) (extra credit) What is the contribution to the kinetic energy from a region where the wave-function is evanescent (e.g., \(x \gt L/2\) )? How does that compare with the potential energy contribution from this same region?

6. Referring to the same potential, U(x) as in the previous problem:
a) What is the (total) energy, \( E_1\) of an electron in this ground state? [hint: Try calculating the total energy directly using the the Schrodinger equation in the well (where U(x)=0). This way, once you understand what to do, you can get \( E_1\) fairly quickly. Also, then you can use that to check your results from problem 5, since K.E. + P.E. = total energy, even in quantum physics.]
b) What is the binding energy of this electron in this state?
c) Thinking about a photoelectric process, what energy and wavelength of photon would it take to free this electron?
d)  (extra credit) Plot the energy of the freed electron as a function of the energy of the incident photon that freed it. (Assume the photon transfers all its energy to the electron.) (Please start both axes scales at 0. I think we can see and learn more that way.)

7. Now let's consider a deeper well. For a 16 eV deep square well that is 0.613 nm wide, I believe that there are exactly 4 bound states and that their energies are:
0.74 eV, 2.95 eV, 6.54 eV and 11.32 eV
Suppose the well has one electron in it, and that electron is in the ground state.
a) What energy of photon would be ideal to excite the electron from the ground state to the 11.32 eV state?
b) Suppose instead we start with the electron in the 1st excited state, and then it drops down to the ground state, emitting a photon. What will the energy of that photon be? What color does that correspond to?
c) Extra: Think about the following: what causes or allows the electron to drop to the lower state?  

8. For the same square well as in the previous problem, suppose the electron starts out in the ground state and then absorbs all the energy of a 16 eV photon.
a) Will that electron still be bound or will it become free?
b) If it is free, what will its kinetic energy be after it absorbs the 16 eV?
c) (extra credit) Sketch a picture illustrating this process.

9.  For the same square well as in the previous problem, now suppose the electron starts out in the ground state and then we gradually change the width of the well until the electron (still in a ground state) has an energy of 3 eV.
Then after that the electron absorbs all the energy of a 15 eV photon.
a) After the electron absorbs the photon, would it be bound or free? What will its kinetic energy be?
b) Think about, and then guess, what well width, more or less, might correspond to an electron ground state energy of 3 eV. Will it be wider or narrower than our original 0.613 nm well, in your opinion.
c) Extra: Think about where the energy might have come from to increase the electron's energy from 0.74 to 3 eV.