Chapter 4, Problem 38P

Using a simple particle-in-a-box model for the multiple bonding in 1,2-butadiene (see Example 4.7) and the $n=2$ wave function for the weakest bound electron, calculate the probability of finding the electron in an 0.1 interval centered midway between the two inner carbon atoms (that is, the center is at $x=2.11\text{Å}$ , so this interval is from $x_1=2.06\text{Å}$ to $x_2=2.16\text{Å}$ ). Then calculate the probability of finding the electron in an 0.1 interval centered midway between an end carbon atom and the carbon atom that is double-bonded to it in the Lewis dot structure. (You may calculate the appropriate integrals or estimate the relevant areas under the curve graphically.) Then recalculate the probabilities by approximating the integral as ${\left|\psi \left(x_0\right)\right|}^2\Delta x$ , where $x_0$ is evaluated in the middle of the range from $x_1$ to $x_2$ . Finally, explain why you can’t approximate the integral needed for Problem 37(a) in this way.