Suppose we put 20 non-interacting quantons into a one-dimensional box of length $L$. The quantons quickly settle into the lowest possible energy states available. Roughly how many times larger will the quantons' total energy be if they are fermions as opposed to bosons? (Choose the closest answer.)
A) About the same
B) About 4 times
C) About 10 times
D) About 40 times
E) About 400 times

Question

Quizplus · Accepted Answer

For bosons, all particles can occupy the lowest energy state, leading to a relatively low total energy. For fermions, due to the Pauli exclusion principle, they must occupy different quantum states, filling up the energy levels. The energy levels in a one-dimensional box are proportional to the square of the quantum number. Summing over the squares of the first 20 integers for fermions and comparing it to the energy of 20 bosons all in the lowest state, the total energy for fermions is roughly 40 times that of bosons.

Suppose We Put 20 Non-Interacting Quantons into a One-Dimensional Box LLL

Suppose We Put 20 Non-Interacting Quantons into a One-Dimensional Box $L$