Final answer:
The probabilities for observing a harmonic oscillator molecule with a specific energy are calculated using the energy levels E = (n + 1/2)ħw and analyzing the distribution of total energy 7.5ħw across five molecules. A combinatorial analysis provides the microstates for each energy level, accounting for the quantum nature of the system.
Step-by-step explanation:
The student's question involves calculating the probabilities of observing a molecule with energy E for five non-interacting harmonic oscillator molecules with total energy Etot = 7.5ħw. The fundamental postulate of statistical mechanics, which states that all accessible microstates are equally probable if the system is in equilibrium, will be used to calculate the probabilities.
The energy levels for a quantum harmonic oscillator are given by E = (n + 1/2)ħw, where n is a non-negative integer. Since the total energy is 7.5ħw for the five molecules, we distribute this energy among them while considering the different ways (microstates) in which each molecule can have an integer number of energy quanta (nħw).
For simplicity, you can consider a few low-energy microstates as examples: If all five molecules are in the ground state (n=0), the total energy would be 2.5ħw which is lower than 7.5ħw, so this state is not possible. Similarly, if one molecule is in the first excited state (n=1, E=1.5ħw) and the others are in the ground state, the total energy would be 4ħw which is also not possible. In a valid state, if one molecule is in the second excited state (n=2, E=2.5ħw) and the others in the ground state, their total energy would exactly match 7.5ħw, and this state would be one of the possible microstates.
The probabilities are determined by counting the number of microstates that correspond to each energy level and then dividing by the total number of microstates. A detailed combinatorial analysis would provide the exact probabilities for each energy level. Note that such distribution usually results in a Boltzmann-type distribution for the probabilities.
Plotting energy versus probability would typically show a peak around certain energy levels, reflecting the most probable energy states for the molecules in the system.