Consider a system with the following single-particle energy-level structure: 0,
ID: 2076068 • Letter: C
Question
Consider a system with the following single-particle energy-level structure: 0, E, E. and 2E. Ignore particle-particle interactions. The system is closed with four identical particles and a total energy of 2E. What is the probability of finding a particle in the ground state if the particles are a) classical particles, b) massive bosons, or c) fermions (s = 1/2) For the fermions, do not ignore spin, however, you may assume the exclusion principle only applies only applies to the spatial, and not the spin, part of the wave function. What does the degenerate first-excited energy level do to the effective "repulsion" or "attraction" towards the ground state as compared with the non-degenerate case? Is the effective "repulsion" or "attraction" increased or decreased? Comment and explain. Your work must make clear your approach for counting states and calculating probabilities. Creating a table in a format similar to the overhead from Week 10, March 29 (found online under Handouts/!figures-from-lecture) is helpful. Note that this problem is asking you to repeat the calculation found on that overhead, but for the case where the first-excited energy level has a degeneracy of two.Explanation / Answer
a)The configuration in which the system has 2E energy and maximum no. of particles in ground state is
3 particles in E=0 state
1 particle in E=2E state
So the probability is 75 percent.
b)The configuration in which the system has 2E energy and maximum no. of particles in ground state is
3 particles in E=0 state
1 particle in E=2E state
So the probability is 75 percent.
c)The configuration in which the system has 2E energy and maximum no. of particles in ground state is
2 particles in E=0
1 particle each in the two E=E states
So the probability is 50 percent
Since there is one particle each in both the first excited energy states, so there is effective repulsion. This repulsion will be less than that in non-degenate case. In non-degenerate state, 2 particles will be in E=0 state and 2 particles in E=E state.