10.5.P6

Light attenuation is the process by which certain frequencies of sunlight (the very low and high frequencies) are absorbed by water as the light travels through it. 

A number of light-absorbing pigments have evolved in fish eyes to allow them to navigate their habitat under a variety of environments and water depths.  In particular, those molecules must allow the fish eye to absorb light that has not already been absorbed by the water above it. In this exercise we seek to understand why these eye pigments absorb in the particular region of the electromagnetic spectrum that they do, and therefore how it is that fish (and animals generally) see in an environment in which the low- and high-frequency portions of the sunlight spectrum are absent.

Let's consider the retinal molecule, which binds to opsin proteins in the eye and forms the basis of animal vision (including fish vision):

1) We'd like to understand how retinal absorbs light, so to start off we need to remember how light corresponds to energy levels in a molecule. Using the energy diagram below, what would happen if an electron fell from the third excited state to the second excited state? What do you think would happen if a photon was absorbed by a molecule represented by this energy level diagram?

This is a general phenomenon that you are familiar with from your science classes. But how are those energy levels determined? A precise answer requires a fair amount of quantum mechanics, but we can obtain a remarkably accurate approximation of the result by considering a fairly simple model for how retinal's electrons behave: the standing wave model.  We'll make use of this model below, and then we'll think about why this model might not be completely accurate.

2) Given the experiment below, where individual photons are shot towards two slits, how can we explain the pattern that shows up on the screen? 

3) It turns out that shooting a beam of electrons gives us a similar result. What does that say about how we treat electrons?

This experiment is compelling evidence that electrons have wavelike properties, much as light does... electrons behave like electron waves!  The retinal molecule's electron waves can be modeled as standing waves, where the standing waves are confined  by the size of the molecule, L.  The first two standing waves for an electron in retinal are shown here (the requirement for these standing waves is that the wave must go to zero at the end points, and either a whole or half number of wavelengths must fit in between):

4)  Draw the 3rd and 4th standing waves for electrons in a molecule of length L.  For each of the drawings, determine the wavelength λ as a function of the molecule length L. Write an expression for wavelength of the n^th standing wave in terms of the length L.

5)  Estimate a value of the length L for retinal, using these empirical facts from chemistry: the average length of a chemical single bond is 0.154 nm and the average length of a chemical double bond is 0.134 nm.   

6) Each of the carbon-carbon double bonds in retinal has two valence electrons that are free to roam around the molecule.  These are the electrons that we have been modeling as standing waves.  Since there are two valence electrons per energy level (remember that from chemistry?!), what is the highest occupied energy level for retinal? What is the first unoccupied level? 

7a) The difference in the energy levels we need to consider is given by E = 1/2 m_ev² = p² /2m. What kind of energy is this? Why does it make sense that we only need to consider this energy?

7b)  The momentum, p, of a standing wave is related to its wavelength by p = h/λ (a result called the deBroglie relationship),  use the above result for the retinal length and your relationship between the wavelength and length of the molecule to determine the energies of the highest occupied energy level and the lowest unoccupied energy level.

8)  What is the energy difference between the highest occupied and lowest unoccupied levels?  Recalling that molecules can only absorb a photon whose energy matches the difference in energy between these levels, what is the energy of the photon most easily absorbed by retinal?  What frequency does this correspond to?  Where in the electromagnetic spectrum does this frequency lie?

9)  Explain why fish would not be able to navigate their surroundings nearly as well if the light-absorbing molecules in their eyes were 10 times smaller or 10 times larger than retinal.

Ben Dreyfus, Ben Geller, & Joe Redish 5/1/12