Homework 4: kinetic energy.

(due Friday by 3 PM) 
Wednesday, 2 pm:  (i just added an extra-credit problem (10) which will be relevant to understanding molecules and crystals.)

Up to this point, we have focused a lot of our attention on wave-functions and degeneracy, and applied that to understanding the some of the periodic table and the geometry of bonding for a few elements (using the 1st-excited states of a -1/r potential).

At this point we can deepen our understanding of quantum physics by turning our attention to energy. Kinetic energy, particularly when associated with confinement, plays a huge role in the quantum world. The interplay and competition between kinetic and potential energy is very important to the phenomena of quantum physics. It can help us understand the size of quantum systems, including the hydrogen atom, the length of chemical bonds and why some materials are insulators instead of metals, as well as some other things. This assignment helps us begin that endeavor.

1. Referring to the enclosed (scanned) notes for clarification as needed,
a) graph exp[-rb/a] as a function of x along the x axis, where rb=sqrt[(x-b)^2+y^2+z^2]
b) do a contour plot to illustrate the nature of exp[-rb/a]

2.
a) graph exp[-r/a] + exp[-rb/a] as a function of x along the x axis.
b) do a contour plot to illustrate the nature of (exp[-r/a] + exp[-rb/a]).

3. 
a) Calculate the expectation value of the kinetic energy (T) for the ground state of a 1D infinite square well.
b) Calculate the expectation value of the potential energy (U) for the ground state of a 1D infinite square well.

4. Note that one can calculate T without reference to any particular potential function. In the world of 1D quantum, consider an electron (of mass m) in a state in the form of a 1D normalized gaussian, i.e., $\psi (x) = (a \sqrt \pi)^{-1/2} e^{-x^2/2a^2}$:
a) calculate T (the expectation value of the kinetic energy) and,
b) graph the expectation value of T as a function of a.

5. On the other hand, potential energy (U) expectation values require both a potential, U(x) and a wave-function, $\psi(x)$.
a) Considering a as a parameter able to take on any positive, real value, calculate the expectation value of U for an electron in a state of the form $\psi (x) = (a \sqrt \pi)^{-1/2} e^{-x^2/2a^2}$,  in a potential $U(x) = (1/2)kx^2$.
b) graph the expectation value of U as a function of a.
c) find the minimum of T + U (expectation values).  At what value of a does that occur? What are the values of T and U at that minimum?

6. For a state that looks like the hydrogen atom electron ground state, but with a allowed to take any value (as in 4 and 5), one can show that T=$\hbar^2/(2ma^2)$ and that
U=$-e^2/a$ (see problem 8 for how one gets this). You can include a $1/(4 \pi \epsilon_o)$ if you prefer that unit system).
a) Show that $\hbar^2/(2ma^2)$ has units of energy,
b) Find the value of a that minimizes E=T + U and the value of T, U and E at that value of a.
c) Are E, T and U positive or negative? Why? (for each one)

7. Consider a 1D finite square well for which the potential is zero everywhere outside the well and -V inside the well. (V, the depth of our attractive well, is a positive number with units of energy.) Assume a width of L, and put the center of the well at x=0.
a) Assuming the ground state is of the form $A cos(kx)$ in the well (A and k undetermined) and $B e^{-\gamma (x-L/2)}$ outside the well, show that you can get a relation between k and $\gamma$ of the form -k*tan(kL/2) =$-\gamma$.
b) Additionally, gamma and k are related since $\hbar^2 k^2/2m = E + V$ and
$\hbar^2 \gamma^2 / (2m) = -E$ (where V is positive and E, the global separation parameter called the energy, is less than zero).
Write $\gamma$ as a function of V and k.
Why is E negative?
c) Anyway you can assume gamma is pretty much constant for small k (why?), and then graph the tan function and 1/k on the same graph to get the actual value of k for the ground state (graphically and approximately). Here is the question:
Is this value of k larger or smaller than the one you would get for an infinite square well of the same width?
d) and even more important: How do you explain that???

8. Extra credit. Calculate the expectation value of T for a hydrogen atom ground state wave function. (take 2 derivatives, integrate...). Do the same for U (i.e., integrate -1/r ...). (This is not actually that difficult as i thought at first since it is only radial derivatives and integrals, i think.)

9. Extra credit/Advanced problem.  Consider an electron in a 1DHO in the mixed state sqrt(1/2)(Psi0(x,t) + Psi1(x,t)).  Calculate and graph U and T (expectation values) as a function of t.

10. extra credit.  For an electron in the ground state of a 1DHO, the calculation of T (the expectation value of the kinetic energy, involves an integral of the product of the ground state wave-function $\psi_o (x)$ and the second derivative of the ground state wave-function (all of which is multiplied by $-\hbar^2/(2m)$.
Carefully sketch the integrand as a function of x. Which regions of x are positive and which are negative? (This will be important.)