Homework 3. due Friday, 2 PM.

1. For an electron in the ground state of a hydrogen atom:
a) calculate the expectation value of $r^2$ . [This requires an integral over all 3D space. Don't forget to include the $4 \pi r^2$ factor and to understand where it comes from.]
b) Calculate the expectation value of the potential energy.
c) Evaluate the expectation value of the potential energy in eV. What is its sign?
d) Think about the relationship between PE, KE and E for this state. Since you probably already know that the energy is -13.6 eV (1 Rydberg), can you guess what the KE must me?



2. Consider the wave-function of an electron in the ground state of a hydrogen atom, $\psi _{100} = ( \pi a^3)^{-1/2} e^{-r/a}$  :
a) For y=z=0 plot this wave-function as a function of x along the x axis.
b) Plot this wave-function as a function of r. (What is the range of r?)


3. Consider the wave-function of an electron in a 1st-excited state of a hydrogen atom called $\psi _{200}$, ( $\psi _{200} = (8 \pi a^3)^{-1/2} (1-r/2a) e^{-r/2a}$)
a) For y=z=0 plot this wave-function as a function of x along the x axis.
b) Plot this wave-function as a function of $r = +(x^2 + y^2 + z^2)^{1/2}$.
c) Where is this wave-function zero?
d) Use a (constant) contour plot to illustrate the nature of this wave-function.

4. Consider the wave-function of an electron in a 1st-excited state of a hydrogen atom $\psi _{21x} = (8 \pi a^3)^{-1/2} (x/2a) e^{-r/2a}$ :
a) For y=z=0 plot this wave-function as a function of x along the x axis.
b) For x=z=0 plot this wave-function as a function of y along the y axis.
c) Where is this wave-function zero? 

5. Use a (constant) contour plot to illustrate the nature of:
a)  $\psi _{21x} = (8 \pi a^3)^{-1/2} (x/2a) e^{-r/2a}$
b) What axes did you use for this ?

6. Use a (constant) contour plots to illustrate the nature of:
a)  $\psi _{21y} = (8 \pi a^3)^{-1/2} (y/2a) e^{-r/2a}$
b)  $( \psi _{21y} + \psi _{21x})/2$ .  Along what direction (in the x-y plane) does the maximum of this wave function lie?

7.  In the hydrogen atom, 1st-excited states can come in many forms. One can add together two, or more, 1st-excited states to make another one. (This is called hybridization.) We include coefficients on these "linear combinations" to maintain normalization.  For example, consider the state:  $(\psi _{21x} - \psi _{200}) / \sqrt{2}$ .
a) For an electron in this state, calculate the expectation value of  x, y and z.
b) Which one is not zero. (Why is that?)
c) what are the units of this expectation value? Is that what you expected?

8. Consider the state:  $(2/3)^{1/2} \psi _{21y} - (1/3)^{1/2} \psi _{200}$ .
a) For this state would the expectation value of x, y or z be non-zero?
b) Whichever one is non-zero, calculate it.
c) Comparing the non-zero expectation value from this problem to the one from the previous problem: which is larger? (extra credit: What do you think might be the origin of that difference?)

9. Consider the two states:  $(1/2)^{1/2} \psi _{21x} -(1/6)^{1/2} \psi _{21y}  - (1/3)^{1/2} \psi _{200}$ and $(2/3)^{1/2} \psi _{21y} - (1/3)^{1/2} \psi _{200}$ .
a) are these states othogonal to each other?
b) are these states normalized? [For these problems you can start from the assumption that { $\psi _{21x}$, $\psi _{21y}$, $\psi _{21z}$ and $\psi _{200}$} are mutually orthogonal and normalized.]
c) (extra credit) Can you find a 3rd state constructed using 1st excited states that is orthogonal to each of these states (and is normalized)? Can you find a 3rd state constructed using only $\psi _{200}$, $\psi _{21x}$ and $\psi _{21y}$ that is orthogonal to each of these states (and is normalized)?

10. a) Calculate the expectation value of the kinetic energy of an electron in the ground state of a hydrogen atom.
b) Evaluate this in eV. Is it positive or negative?