Electrons on energy levels - a review of microcanonical statistical thermodynamics: We consider a system composed of 4 electrons, which can occupy quantized (i.e. discrete) energy levels. The available energy levels are equidistant: the constant spacing between levels is E0=0. Here we take as the reference of energy the lowest level (which amounts to neglecting the ground-state energy). The diagram of possible energy levels is sketched below. Each "rung" in the energy diagram ladder is a state that can accommodate at most 2 electrons (i.e. there can be up to 2 electrons per chemical orbital). One example of a possible configuration is shown on the right side. In this problem, we assume that the system was prepared such that its total energy is 6E0. You are asked about possible distributions of the electrons on the energy levels and the associated statistics (microcanonical ensemble). In a) and b) we are taking two steps to enumerate the number of possible microstates. a) (10 points) Conservation of energy: give all the possible configurations of electrons (i.e. the possible ways to distribute the 4 electrons that are compatible with the total energy). Keep in mind that at most 2 electrons can occupy the same state. You should find FIVE additional types of configurations, in addition to the example one. Here, you do not need to keep track of the spin, only consider which energy levels the electrons occupy (so you could draw electrons as balls with no arrow on them). b) (10 points) Now, we explicitly consider that electrons carry a spin (arrow up or down). How many microstates are there, including all the different spin states? What is the probability of each microstate (microcanonical picture)? To find the answers, remember these physical chemistry rules: if two electrons occupy the same level then they need to have opposite spins. Also, if a level is occupied by two electrons of opposite spins, then it does not matter which one is spin-up and which one is spin-down, since electrons are indistinguishable particles. But if a level is occupied is occupied by a single electron, its spin orientation matters: it could be up or down (counts as two separate cases). c) (5 points) We are interested in the total spin of the system, projected along the z (vertical) axis: Sz. This quantity is simply the sum of spins of individual electrons, where a spin-up electron contributes +1 to Sz and a spin-down electron contributes -1 to Sz. Based on possible configurations found in a), what are the possible values for Sz and what are their probabilities