Unit 58: Wavefunctions & Probability Density
Practice Problems
Note: problem difficulty is ranked using a star system.
(*) One-star problems are fundamental to the unit, and can be done relatively quickly. Use these problems to introduce yourself to the material.
(**) Two-star problems are more difficult, and require an understanding of one or two key concepts. Use these problems to test your understanding of the material.
(***) Three-star problems are the most difficult, and require some creative thinking in addition to a deep familiarity with multiple key concepts. Use these problems to challenge yourself; if you can complete one of these, you’re on your way to mastering the material.
*Q58.1) Consider a molecule of 1,3,5,7,9-Decapentaene (shown below).
C = C - C = C - C = C - C = C - C = C
You may ignore any bond-angle effects, and use a carbon-carbon bond length of 0.15nm.
a) Sketch the wavefunctions for n=1, n=2, and n=3 π-electrons in this molecule.
b) Sketch the probability density functions for n=1, n=2, and n=3 π-electrons in this molecule.
c) At what positions along the molecule are n=3 π-electrons most likely to be found?
d) At what positions along the molecule are n=3 π-electrons least likely to be found?
e) What’s the wavelength of an n=3 π-electron in this molecule?
f) What’s the maximum probability density of an n=3 π-electron in this molecule?
g) What’s the probability of finding an n=3 electron within a region of width 0.02nm centered at the 5th carbon atom?
h) What’s the probability of finding an n=3 somewhere between the 4th and 7th carbon atoms?