Homework 4. Due Monday Feb 4 (3 PM).

(energy, oscillators, hybrids and L) 
Energy:
1. a) Calculate the expectation value of 1/r for an electron in the state \( \psi_{21z} \).
b) Use that to obtain the expectation value of the potential energy for an electron in this state.
c) evaluate that in eV.
d) using your knowledge of the total energy, infer the kinetic energy for an electron in this state.

2. a) Do the same thing for an electron in the state \( \psi_{200} \).
b) Summarize your results in one or two cogent sentences. Include a comparison to what you already know about the ground state.


Harmonic Oscillator:
3. For a 2-dimensional harmonic oscillator, the most essential part of two first excited states can be written as:
$$ x e^{-(x^2+y^2)/(2a^2)} $$ $$ y e^{-(x^2+y^2)/(2a^2)} $$  (These span the space of first excited states.)
a) Do a contour graph of each of these.
b) normalize them.
c) Can you make off-center hybrid states from these states? why or why not? (an off-center state is one for which the expectation value of x or y is non-zero.)

4. For a 3-dimensional harmonic oscillator the most essential part of 3 1st-excited states (unnormalized) can be written in the form:
$$ x e^{-(x^2+y^2+z^2)/(2a^2)} $$ $$ y e^{-(x^2+y^2+z^2)/(2a^2)} $$ $$ z e^{-(x^2+y^2+z^2)/(2a^2)} $$   (these also span)
a) Can you make off-center hybrid states with these? why or why not? (an off-center state is one for which the expectation value of x, y or z is non-zero.)
b) (extra) Think about and discuss the difference between 3D harmonic oscillator 1st-excited states and H-atom 1st excited states in this regard. What are the implications?

 Hybrids (of hydrogenic 1st excited states):
5. This problem is about constructing \(sp^2 \) hybrid states from the 1st excited states {\( \psi_{21x} \), \( \psi_{21y} \), \( \psi_{21z} \), \( \psi_{200} \)}. (These states provide a basis and are mutually orthogonal and normalized.)
a) Consider the state \( \frac {\sqrt{2}} {\sqrt{3}} \psi_{21x} - \frac {1} {\sqrt{3}} \psi_{200} \). In what direction does this wave-function "point"?
b) Starting with this state as given, construct the other  \(sp^2 \) hybrid states. [ \( \psi_{21z} \) remains un-hybridized, so really you have two hybrid states to figure out, and those are in the x-y plane.]
c) sketch a picture of Benzene (C6H6) or ethylene (C2H4) and show how your \(sp^2 \) wave-functions relate to that.

6. This problem is about constructing \(sp^3 \) hybrid states from the 1st excited states {\( \psi_{21x} \), \( \psi_{21y} \), \( \psi_{21z} \), \( \psi_{200} \)}.
a) Let's start with the state \( \frac {\sqrt{3}} {\sqrt{4}} \psi_{21z} - \frac {1} {\sqrt{4}} \psi_{200} \). In what direction does this wave-function "point"?
b) Starting with this state as given, think about and describe the other 3  \(sp^3 \) hybrid states. (There are a total of 4.)
c) Construct the other 3  \(sp^3 \) hybrid states. [Hint: they each have the same coefficient on their \( \psi_{200} \) state.]
d) The coefficient on the \( \psi_{200} \) state is smaller than for the sp2 states. Why do you think that is?
 e) (extra work) Sketch a picture of some molecule or structure exhibiting \(sp^3 \) wavefunctions and think about how your wavefunctions relate to that.

7. When you are calculating the expectation value of x, y, or z for a hybrid state (an off-center hybrid), one can show that all non-zero integrals have integrands of the form:
\( x \psi_{21x} \psi_{200} \),
\( y \psi_{21y} \psi_{200} \), or
\( z \psi_{21z} \psi_{200} \).
a) list the \( \phi \) integrands that arise in this context. (List just the integrand, not the integral, e.g., \( sin^2( \phi ) \)is an integrand.
b) List each \( \theta \) integrand that arises in integrals of this form (same format).
c) (extra task) Also list some \(theta \) and \( \phi \) integrands that are zero, that arise when calculating expectation value of x, y, or z for hybrid (1st-excited) states.

Angular momentum:
8. Calculate
$$ - \hbar^2 /sin( \theta) \frac {\partial} {\partial \theta} (sin( \theta ) \frac {\partial \psi_{21z}} {\partial \theta}) $$
 a) Do you get back something proportional to \( \psi_{21z} \) ? if so, what is the proportionality factor?
b) The "operator" \( - \hbar^2 [\frac {1} {sin( \theta)} \frac {\partial} {\partial \theta} (sin( \theta ) \frac {\partial } {\partial \theta}) + \frac {1}{sin^{2} ( \theta)} \frac {\partial^2} {\partial \phi^2}]  \) is called \( L^2 \). Is \( \psi_{21z} \) an eigenstate of \( L^2 \) ?
c) (extra work) Is \( \psi_{21x} \) an eigenstate of \( L^2 \) ? Try it. Is this hard?
(There is a derivation of \(L^2\) in appendix G of Harris, Modern Physics.)

9.  The z-component of the angular momentum operator can be written \(L_z = - i \hbar \frac {\partial} {\partial \phi}\).
a) Show that when you operate on \( \psi_{21x} \) with this operator, that you get a result proportional to  \( \psi_{21y} \).  (That is, calculate \( - i \hbar \frac {\partial \psi_{21x}} {\partial \phi}\) and examine what you get...)
b) Similarly, show that when you operate on \( \psi_{21y} \) with \( L_z \) you get a result proportional to  \( \psi_{21x} \).
c) Additionally, show that  \( \psi_{21x} + i \psi_{21y}  \) is an eigenstate of  \( L_z \). What is its eigenvalue?
d)  What do you get when you "operate"  the operator \( L_z \) on the state  \( \psi_{21z} \). What does that imply about \( \psi_{21z} \) ?
e) (Extra work) Explore the relationship between \( xp_y - yp_x \) and \(- i \hbar \frac {\partial} {\partial \phi}\)