Aquantum dot is modeled as a particle (exciton) of reduced mass equal to the mass of the electron confined in a cubic 3-d box of dimension l40nm on a side: (a) sketch the first four sharp energies i. e., draw an energy level diagram. label each energy with the appropriate quantum number triplet, nz; ny; n. how many different solutions to the time-independent schrödinger equation exist at each energy? (b) the confinement energy is 1.9 mev. what are the quantum numbers nai ny; n? are they uniquely determined? (c) quantum dots can be synthesised in a range of sizes, by changing the reaction conditions. if the reaction temperature is increased so that the quantum dots produced are 10% smaller in all three dimensions, how will this impact on the energy spectrum from part (a)? (d) write down a possible wavefunction for the confined particle. what is the probability density for finding the particle at position (x; y, z) = (0: 67l; 0: 67l: 0: 67l)?