Homework 5: momentum, confinement and kinetic energy.

due Friday, Feb 8.
(corrections: problems 3 & 7, Tuesday evening (see comments))
(edited: Sunday Feb 3, intro and problem 8)
(Please report any typos you find and ask about anything that is unclear, strange, etc.)

This is probably our most important homework so far in this class. I think it will be pivotal in helping develop your understanding of key aspects of quantum physics which will allow us to make the transition from describing atoms and other single-well systems to the description of molecules,  semiconductors and other quantum materials. Feel free to ask questions and discuss the problems. These are hard problems and I think it is helpful to work with actual groups and people, and to engage in questions and discussion here.

Problems 1-4 are related to confinement, momentum and kinetic energy. As you know, the Heisenberg relation is often expressed as the inequality $\Delta x \Delta p \ge \hbar / 2$. In the first 4 problems we will do some calculations of $\Delta x$ and $\Delta p$ (and some closely related quantities) for specific systems and states. This will provide a chance to examine the Heisenberg inequality from a physics point-of-view, which is somewhat different from perspectives that are pervasive in popular culture (which tend to fixate on the word "uncertainty"). Uncertainty is a complex word with many possible meanings. Inequalities, such as the one above, refer to the precision with which one can localize a wave-packet, and the need for short wave-lengths in doing so. Such relationships are present in many wave theories, string, E&M, water ..., in which one can envision deconstructing a localized packet into components of precise (well-defined) wavelength.

Problems 5-7 all focus on kinetic energy integrands, especially distinguishing regions of space that contribute negatively to the kinetic energy from the regions which contribute positively to KE. This is a novel and important aspect of quantum mechanics. Another perhaps novel thing is that all the kinetic energies in 5-7 are independent of time.  However, this is not generally the case for KE in quantum physics.

Problem 8 provides a relatively simple example of a state (wave-function) in which KE and PE of an electron vary (in a complementary way) as a function of time. I think if you take the time to do 8, it provides a perspective that allows the whole thing to make more sense. Feel free to start with part b), which is mathematically a little easier than a).

1. a) Calculate $\Delta x$ and  $\Delta p$ for an electron in the ground state of a harmonic oscillator.
    b) What is the product $\Delta x \Delta p$ for an electron in this state? Does it depend on time?

2. a) Calculate $\Delta x$ and  $\Delta p$ for an electron in the ground state of an infinite square well (of width L).
    b) What is the product $\Delta x \Delta p$ for an electron in this state?

3. Show that the Heisenberg relation can be recast as a relationship between kinetic energy and confinement, namely,
$\bar{T} \geq \hbar^2/(4 m (\Delta x)^2$, aaarrrrg, this is incorrect.
$\bar{T} \geq \hbar^2/(8 m (\Delta x)^2$, <---- this one is corrrect i think.
where $\bar{T}$ is an expectation value of kinetic energy and $\Delta x = \sqrt{\bar{x^2}-\bar{x}^2}$ is a length scale which provides a measure of the degree of confinement of an electron.

4. Write $\bar{T}$ in terms of $(\Delta x)^2$, m and $\hbar$ for:
a) an electron in the ground state of a harmonic oscillator.
b) an electron in the ground state of an infinite square well.
(these are different, right?)

5. a) Graph the integrand from the integral that arises in the calculation of $\bar{T}$ for an electron in the ground state of a harmonic oscillator.
 b) Although  $\bar{T}$ is positive, there are regions of x that contribute negatively to the integral (regions that diminish $\bar{T}$).  Where are there? (That is, over what range(s) of x is the integrand negative?)
c) Indicate those regions in your graph. Do you notice anything about the nature of the curvature of $\psi (x)$ for those regions?

6. a) Graph the integrand from the integral that arises in the calculation of $\bar{T}$ for an electron in the ground state of a finite square well. (Note: For this problem you are not expected to calculate T or to know the exact state for this problem. Please use the form of the ground state that was provided in problem 7 of HW 4. I think you can do this entire problem using that form (and without evaluating A or B).)
 b) Although  $\bar{T}$ is positive, there are regions of x that contribute negatively to the integral (regions that diminish $\bar{T}$).  Where are there? (That is, over what range(s) of x is the integrand negative?)
c) Indicate those regions in your graph. Do you notice anything about the nature of the curvature of $\psi (x)$ for those regions?

7.  In this problem we would like to sketch the integrand associated with the $\bar{T}$ calculation for a double square-well focusing in particular on the nature of the contribution to $\bar{T}$ that arises from the region between the two wells. A double square-well refers to two finite square wells. In this case, let's put the origin of coordinates between the wells, define a center-to-center distance of b, and the width of the wells as L. In equations:
$U(x) = -V$, for (-b/2) - (L/2) $\leq x \leq$ (-b/2) + (L/2),
$U(x) = -V$, for (b/2) - (L/2) $\leq x \leq$ (b/2) + (L/2),
$U(x) = 0$, elsewhere.
(In this case, the two wells are identical in depth (measured in energy) and width (L). We assume that b is greater than L, so that the wells are not overlapping and that b is not too large,  so that the two wells are close to each other. (We will learn how to define what "close" means. That is an important question! (thanks for asking).

Additionally let's use the following notation for the ground state:
$\psi(x) =  A cos(k(x-b/2))$, for (b/2) - (L/2) $\leq x \leq$ (b/2) + (L/2),
$\psi(x) =  B e^{-\gamma (x-(b+L)/2)}$, for $x \geq (b+L)/2)$,
$\psi(x) =  C cosh(\gamma x)$, for (-b/2) + (L/2) $\leq x \leq$ (b/2) - (L/2),
Where the A cos( ) term is for the wave-function within the region of the right-hand-well, which is centered at b/2, I believe; the $B e^{-( )}$ term is for the region beyond the right-hand-well, and the very important C cosh( ) term is for the region between the two wells

 a) graph the potential U(x) as a function of x (for b = 2L approximately).
 b) graph the ground state wave-function $\psi (x)$ as a function of x. (I would suggest using an assumed $\gamma$ of roughly 3/b.)
 c) calculate and then graph the second derivative of the wave function for these regions (leaving A, B and C undetermined)
 d) graph the integrand of the kinetic energy calculation for this double-well system,
$\psi (x) (-d^2 \psi (x) / dx^2)$ for these regions.
 e) What is the sign of the contribution to $\bar{T}$ from the region between the wells? Why might this be relevant to understanding molecular bonding? Thinking about K.E. and confinement, what does it signify?

8.  Advanced. (I would recommend reading the whole problem and then starting with the calculation of part b. I think that is a lot less difficult than calculating $\bar{T}$ for this mixed state.) All of the kinetic energy expectation values we have calculated so far have been "constants of motion", that is, not dependent on time. This is a different from what you may be used to, since in a classical harmonic oscillator, or a wave on a string, etc., we expect the kinetic and potential energies to oscillate with time. This can happen in QM also, however, just not for the pure states (energy eigenstates) that we have been studying. 
a) Show that for an electron in a harmonic oscillator in the mixed state that is an equal mix of the ground state and 2nd excited state, the kinetic and potential energy each oscillate as a function of time. What is the period of the oscillation? How does it relate to an oscillation in the size of the wave-function?
e.g., at t=0, $\psi(x) = (\psi_o (x) - \psi_2 (x))/\sqrt{2}$
Start here ==>  b) Calculate $\bar{x^2}$ as a function of time for an electron in this mixed state. What is its smallest value? What is is largest value? Graph it (vs t). How is $\bar{U}$ related to $\bar{x^2}$? How would you guess that $\bar{T}$ might be related to $\bar{U}$?  (Feel free to discuss here. Is there a simple relationship between $\bar{T}$ and $\bar{U}$? This is not obvious so anything you say or ask will be good.)