CHM 3411, Dr.
Chatfield, Fall 2009
Problem Set 3
Due Friday, Jan.
30
Suggested (b) Exercises from
The first question deals with orthonormality (the answer is simple if you see it). The next four questions explore the particle in a box. The wave functions are given in the text. We will cover the material for questions 2-4 in class on Monday, but you should be able to handle problem 2 before that.
1. As discussed in class, the solutions (i.e. wavefunctions) to the time independent Schrödinger equation form a complete orthonormal set of functions. One consequence is that an arbitrary function f(x) may be writeen as a linear combination of these wavefunctions:
where ci are constants. Using the general properties of completeness and orthonormality, show that the constants ci are given by the expression:
2. Calculate the probability for finding a particle in the middle third of a 1-D box for each of the three lowest-energy states (n = 1, 2, 3). Sketch the wave function and the probability density as a function of x for each case, and show qualitatively (be relating the probability to the corresponding area on the probability density graph) that the results of your calculation make sense.
3. Calculate the degeneracies of the first four energy levels of a particle in a cubic 3-D box. (Note: the degeneracy is the number of different wave functions (i.e. states) having the same energy level. We often use the letter “g” to represent degeneracy.)
4. As explained on p. 281 of the text, the p-electrons in a system of conjugated double bonds can be modeled qualitatively as a particle-in-a-box problem. This is because the electrons are free to roam the entire length of the conjugated chain. Carotenoids, a class of molecules including b-carotene, are colorful because they absorb in the visible region of the spectrum due to extended conjugation. Three carotenoids are shown below:
Note that b, a and e-carotene have 11, 10 and 9 conjugated double bonds, respectively. The wavelengths at which b, a and e-carotene absorb visible light are 450, 445 and 440 nm, respectively. Use a 1-D particle-in-a-box model to calculate the wavelengths at which these molecules absorb radiation. Note that one should not expect exact results with such a simple model. In fact, the particle-in-a-box solutions will differ from the experimental results by a factor of between 2 and 3. They will, though, correctly capture the trends.
Hint: The length of the box can be determined from average lengths for single and double carbon-carbon bonds (1.450 and 1.335 Angstroms, respectively). Note that the box actually has a zig-zag shape, but it is still 1-D, so we can apply the particle-in-a-box solutions we derived earlier. Also note that the Pauli exclusion principle applies, that is, only two electrons can occupy a given orbital. Thus, to create the ground state, count the number of p-electrons and fill up the energy levels with two electrons each, beginning with the lowest energy level (these energy levels are non-degenerate). The first excited state is created by exciting an electron from the highest occupied level to the lowest unoccupied level. This happens when light of the appropriate frequency is absorbed by the electron, i.e.
where n is the quantum number of the highest occupied level in the ground state.
5. Atkins, Problem 9.8.
6. Atkins, Problem 9.9.