Cormorant’s mass is 2.5 kg and he is flying at a constant speed of 2.0 ms-1 in the tube:
① What would his kinetic energy be under these circumstances?
② The energy of a particle in the box is entirely kinetic, since the potential energy is zero everywhere within the box. We deduced in section 6.5 that this energy is given by:
Deduce the value of n that would be associated with Cormorant’s wavefunction.
③ One of Cormorant’s concerns is that – depending on which quantum level he is in - he might be more likely to be found in some parts of the box than others. This seems completely inconsistent with his notion of free movement within the box. Should your answer to ② reassure him on this point? (Prince Louis will help you with this one)
④ Another concern is that the kinetic energy quantisation seems to mean that he would be able to fly at this speed or that speed but not at any speed in between. Can you reassure him again?
⑤ Let's now compare Cormorant's experience with that of an electron in a molecule. It's a very different scale: Cormorant is finding it hard to empathise.
Specifically, we'll revisit the buta-1,3-diene system that we investigated in Question 6-11. We would like to ask, for one of those delocalised electrons in butadiene, what kind of quantum level it is likely to be found in (in Question 6-11, we assumed it would be in the ground state but was that a valid assumption?).
Statistical thermodynamics – a whole other branch of physics, and a can of worms that you’ll have to save for another day – teaches that if you have a population of particles in thermal equilibrium, the average energy a particle will have is ½ kBT for each different way in which it can move, where kB is the Boltzmann constant and T is the absolute temperature (in K). Now this is of limited applicability here because we have don’t have much of a population: we have precisely one electron. However it is still useful to give us a rough measure of the range of energy levels that will be accessible to an electron in a molecule at a given temperature.
In Question 6-11 we calculated that the energy needed to promote an electron from the n=1 state to n=2 was 9.6 x 10-19 J. Compare this with ½ kBT and hence consider (qualitatively) the likelihood that an electron can reach an excited state at around room temperature. (For this you’ll need to know that kB = 1.38 x 10-23 JK-1).
