HW 3. FSWs, probablity density, evanescent wave-functions, energies...Solutions added 4-20

In this HW set, we will begin to explore and learn about the nature and meaning of the wave function and why \(\psi^{*} \psi\) is called the probability density.
If you are interested in that, I would recommend planning a lot of time to work on this. Working with other students in this class, who are going through the same thing you are, could be helpful. I think that ideally you could want to work on this over a few days to give you time to mull things over.

Due Friday, April 19. There are 10 problems. 
Please be sure to see problem 10 at the bottom. Also, the section "Additional things you could work on and think about to learn more" is important to look at and work on when you have time, even though you do not have to hand that in.


For a finite square well centered around zero, the spatial part of even energy-eigenstate wave-functions can be written in the form:
\(\psi = A cos(kx)\)     inside the well and
\(\psi = B e^{-b (x-L/2)}\)     outside the well on the positive x side.

For odd states it is:
\(\psi = A sin(kx)\)     inside the well and
\(\psi = B e^{-b (x-L/2)}\)     outside the well on the positive x side.

Some nomenclature: k and b are wave-function parameters; A and B are normalization parameters. A and B are not easy to determine and you will have to use approximate numerical methods. (The width of the well, L, the depth of the well, U, and m and \(\hbar\) are input parameters.) ISW stands for infinite square well. FSW stands for finite square well.

To solve for the wave-function parameters one uses boundary conditions at one edge of the well (on the right typically), along with the (time-indendent) Schrodinger equation, the input parameters, and the fact that energy, E, is a global separation parameter and therefore must have the same value inside and outside the well for a particular state. You can solve these problems using just the Schrodinger equation and boundary conditions, however, you will need to do numerical computation as they can't be done is "closed form".  For all results involving a length or energy scale, please use only eV and nm.

1. Energy comparison. Consider a finite square well (FSW) potential centered around x=0 with a width of 0.613 nm. Suppose the potential function is zero inside the well and 3 eV outside the well.
a) Using boundary conditions and the Schrodinger eqn, find the total energy of an electron in the ground state of this well. How does this ground state energy compare with the ground state energy of an electron in an infinite square well of the same width?
b) What is the energy of an electron in the first excited state? How does that compare with the energy of an electron the first excited state of an infinite square well of the same width?
c) How many bound states does this particular attractive well have?
(Spend some time thinking about the difference between the states, energies and wave-function parameters for this finite square well and an infinite square well of the same width.)
Commentary on problem 1. added Wed, 7 AM.
Here is an interesting question I got from a student:

Student: I've been doing homework 3 and I just want to clarify what we're expected to do for problem 1.  For example, for 1a it asks to find the total energy in the ground state of the finite well. So the general process would be to: 

1) find k and b using boundary conditions and graphing

2)Find A and B by utilizing normalization

3)calculate the expectation values of KE and PE inside and outside the well (6 integrals total)

4) Add up the expectation values.

Is this the general method or is there an easier way? This seems to be a lot of work just for one part of one problem. 

Prof: There is a much easier way to do problem 1.  Inside the well there is a relationship between k and energy which is, for the ground state: \(\hbar^2 k_1^2/(2m) = E_1\). You can thus find the total energy of the state from \(k_1\), which you obtain numerically using a method like that in the video on Square Well Boundary Conditions. The more difficult calculations you describe are needed to dissect and analyze the different parts of the energy (and thereby gain a deeper understanding of the origin and nature of an energy-eigenstate wave-function).  Getting the total energy is much less difficult and you do not need the normalization parameters, A and B, to get just the total energy.

Also, when trying to find the total energy, are you supposed to add every energy expectation value you find (all 6)? If I have an expectation value for the kinetic energy in all 3 regions and a potential energy in all 3 regions, would I add up all 6 of those values to get the total energy? 

Prof: You are totally correct that there are 6 parts, and yes you do add them to get the total energy. However, the outside ones are equal by symmetry, and one of the P.E. parts will be zero usually. But to find the total energy, you do not need to do all that.

2. Wave function parameters. Consider an electron in a square well that is 2 nm wide.
a) What are the values of the wave-function parameter k for the ground state, 1st excited state and 2nd excited state in the limit where the well becomes infinitely deep?
b) What would the energy be of an electron in the ground state, 1st excited state and 2nd excited state, respectively, in the limit where the well becomes infinitely deep?

3. Wave function parameters.  Consider an electron in a square well that is 2 nm wide and 0.3 eV deep (centered at x=0), with V(x)=0 outside the well and V(x) = -0.3 eV inside the well.
a) Using boundary conditions and the Schrodinger eqn, find the wave-function parameters for the ground state of this well.
b) Using boundary conditions and the Schrodinger eqn, find the wave-function parameters for the 1st excited state of this well.
c) Take some time to think about the difference between these wave function parameters and those of an infinite square well of the same width. Discuss your thoughts about that here.
d) What would the energy be of an electron in the ground state, 1st excited state and 2nd excited state, respectively.
e) What energy of photon would be required to excite an electron from the ground state to the 1st excited state?  What energy of photon would be emitted if an electron dropped from the 1st excited state to the ground state?

4. Do the states of a finite square well tend to have lower energy than the states of an infinite square well? If so, why do you think that is. (For this comparison would it make sense to align the potential energy function at the bottom of the well?)

5. Ground state wave-function and normalization. (This problem is not easy and I think you may need a few days to think about it in addition to quite a few hours of work in order to actually successfully complete it. It involves numerical calculations that are more difficult and time consuming than the problems above.)
Consider an electron in a square well that is 2 nm wide and 0.3 eV deep (centered at x=0).
a) Using boundary conditions, normalization integrals, and your prior results, find the normalized ground state wave function. That is, find actual values for k, b, A and B accurate to 3 significant figures, ideally in units involving nm.
b) What is the value of this wave-function at x=0? (The parameter A.)  How does that compare with an infinite square well of the same width?
c) What is the value of this wave-function at x= 1nm? (The parameter B, I think.)  How does that compare with an infinite square well of the same width?
d) What is the value of this wave-function at x=2nm?  How does that compare with an infinite square well of the same width?
e) Summarize your results. In what regions if the FSQ wave-function smaller than the ISQ wave-function? In what regions is it larger?

6. First excited state wave-function and normalization. Do the same calculations for the first excited state.
a) What is the value of the first-excited state wave-function at x=2nm?
b) What is the value of the ground state wave-function at x=2nm?
c) How do they compare? Which is larger?
d) Compare and discuss the length scales associated with the evanescent part of the wave function for the ground state and 1st excited state, respectively. (That is, the part of the wave-function that extends outside the well. Since it is pretty much the same on both sides, you can focus on just the right side. Would that length scale be 1/b? Why or why not? What are the units of b?)
e) Which state extends further outside the well?

7. Probabilty Density. Consider an electron in a square well that is 2 nm wide and 0.3 eV deep (centered at x=0).
a) The integral of the probability density over all space is 1. For the ground state, what percentage of that integral lies inside the well? What percentage lies outside the well?
b) The integral of the probability density over all space is 1. For the 1st excited state, what percentage of that integral lies inside the well? What percentage lies outside the well?
c) These calculations provides a measure of the degree to which the electron wave-function extends outside the well into what is called a classically forbidden region. Which electron wave function extends into the forbidden region more?

8. Potential Energy. Consider a square well that is 2 nm wide and 0.3 eV deep (centered at x=0).
a) Calculate the expectation value of the potential energy for an electron in the ground state. How does that compare with the expectation value of the P.E. for an electron in an ISW of the same width?
b) What can you infer regarding the value of the expectation value of the kinetic energy for this electron? How does its value in eV compare with that of an electron in the ground state of an infinite square well of the same width? Is it larger or smaller? What are your thoughts on why that might be?
c) Comparing this FSW to an ISW of the same width: did the P.E. go up or down? Did the K.E. go up or down? Did the total energy go up or down? Think about this and discuss and explain here.

9. Kinetic Energy. Consider an electron in the ground state of a square well that is 2 nm wide and 0.3 eV deep (centered at x=0).  Let's think more about the kinetic energy. The expectation value of the kinetic energy involves 3 integrals, one inside the well and two outside the well. 
a) Calculate just the integral outside the well on the right. (This is an pretty easy integration once you have B and b.) What number, in eV, do you get from this integration?  Is it positive or negative?
b) Assuming that the integral outside the well on the other side will yield the same value, which is a really good assumption, what is the influence of these evanescent regions on the expectation value of the kinetic energy?
c) Considering your results from problem 8, what fraction of the reduction of the kinetic energy expectation value is associated with the integrations outside the well.
d) Why is the kinetic energy lower anyway??  (of the electron in the ground state of an FSW compared to an electron in the ground state of an ISW of the same width).

10. Attractive delta function.  Consider a potential of the form:
\(V(x) = -\alpha \delta(x)\)
\(\delta(x)\) is a Dirac-delta function and \(\alpha\) is positive and real.
a) Show that the ground state of this potential takes the form:
\(\psi_1 (x) = a^{-1/2} e^{-|x|/a}\)
That is, does this state satisfy the Schrodinger eqn and the unusual boundary condition at x=0?
b) Is this state normalized? (Please send me an email asap if not.)
c) Calculate the expectation value of the potential energy for an electron in this state.
d) Calculate the size of an electron in this state.
e) How does that size depend on \(\alpha\)?  What are the units of \(\alpha\)?
f) Graph the size vs \(\alpha\).
g) Why is a kinetic energy expectation value calculation difficult for an electron in this state? What is the issue with that?

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Additional things you could work on and think about to learn more. I highly recommend that you look at these.

Wave function parameters. Consider an electron in a square well that is 1 nm wide and 1 eV deep, with an infinite potential wall at x=0 on the left. That is:
V=infinity for x<0;  V=-1.0 eV for 0<x<1 nm;   V=0 eV for x>1 nm.
a) What are the values of the wave-function parameter k for the ground state and 1st excited state for specifically this potential? How do they compare with the case of an infinite square well that is 1 nm wide?(3 sig figs is good.)
b) What is the energy of an electron in the ground state?
c) sketch a plot of the ground state wave-function.
d) Obtain the normalization parameters for the gs wave-function accurate to like 2 or 3 sig figs.
e) Calculate the expectation values of the kinetic energy and potential energy for an electron in the ground state of this potential.
f) Calculate the expectation value of x for an electron in the ground state of this potential.
g) Do the same things for an electron in the first excited state.

Solutions: (added 4-20)
HW3 solutions