Schrödinger's Cormorant

(6-11) ① Here is some relevant information:

If the number of double bonds in the conjugated system is q, that means we have 2q electrons in the system. For reasons that we won't get into at the moment, only 2 electrons can be in each eigenstate of the system.
We can define the building block of a conjugated system as comprising a pair of carbon atoms with one double bond and one single bond, as shown below. We can then define the length of our box (L) in terms of the length of this basic unit (j):

(apologies about the bond angles, or lack thereof, in this picture: we're taking a very simplistic view here, so we'll pretend the molecule is linear, just for simplicity).

An electron can be promoted from the highest energy occupied state to the lowest energy unoccupied state, by absorbing energy in the photon of an appropriate wavelength.
When we solved the Schrödinger equation for the particle in a 1-dimensional box, in section 6.5, we derived the energy expression, for a particle in the nth energy level:

Assuming that the electrons are in the n=1 state to start with, see if you can use all this information to predict what happens to the energy required to promote an electron, and hence to the wavelength of the radiation that can be absorbed by a conjugated system, as the number of pairs of carbon atoms increases.

② Buta-1,3-diene has the structure CH2=CH-CH=CH2 . See if you can now calculate the predicted energy needed to promote an electron in buta-1,3-diene to its lowest energy excited state. Hence deduce the wavelength of radiation which will provide photons of the appropriate wavelength. You’ll need some numbers to work with:

Typical C-C bond length in a polyene: 1.46 x 10-10m

Typical C=C bond length in a polyene: 1.34 x 10-10m

Planck’s constant: 6.63 x 10-34 Js

Mass of an electron: 9.11 x 10-31 kg

Do you expect this compound to be coloured?

③ Tomatoes are red because they contain a compound called lycopene. Its structure is:

You can see that the middle bit of the molecule is a conjugated system which we can therefore model using the particle in a box. First, use the result from ① to explain qualitatively why this compound is coloured. Then calculate the predicted absorption wavelength for this molecule.

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