Question

Consider a system of five particles, inside a container where the allowed energy levels are nondegenerate and evenly spaced. For instance, the particles could be trapped in a one-dimensional harmonic oscillator potential. In this problem you will consider the allowed states for this system, depending on whether the particles are identical fermions, identical bosons, or distinguishable particles.

(a) Describe the ground state of this system, for each of these three cases.

(b) Suppose that the system has one unit of energy (above the ground state). Describe the allowed states of the system, for each of the three cases. How many possible system states are there in each case?

(c) Repeat part (b) for two units of energy and for three units of energy.

(d) Suppose that the temperature of this system is low, so that the total energy is low (though not necessarily zero). In what way will the behavior of the bosonic system differ from that of the system of distinguishable particles? Discuss.

Short answer

(a) The ground state of this system is.

(b) The are five possible system states in each case.

(c) For two-unit of energy the graph is

and for three-unit of energy is

.

(d) The way that the behavior of the bosonic system differs from that of the system of distinguishable particles is discussed below.

Step-by-step solution

Part (a) Step 1: Given Information

We have to describe the ground state of the system, for each of these three cases.

Part (a) Step 2: Simplify

As bosons do not follow the Pauli exclusion principle, particles in the ground state on the same level are distinguishable, but if they are fermions, each one will occupy a level starting from the lowest level, resulting in something like this:

Part (b) Step 1: Given Information

We have to describe the allowed states of the system, for each of the three cases and find the possible system states in each case.

Part (b) Step 2: Simplify

Consider a particle that has been promoted to the second lowest level, one energy unit above the ground state. There is only one way to do this for identifiable particles or bosons, which is to promote one particle from the lowest level to the second lowest level. There are five ways to promote indistinguishable particles or fermions, and the particle is promoted from the fifth level to the sixth level, as represented graphically:

Part (c) Step 1: Given Information

We have to repeat part (b) for two units of energy and for three units of energy.

Part (c) Step 2: Simplify

Consider a particle promoted to the second lowest level with two energy units above the ground state. We can promote one particle up to two energy levels, leaving four particles in the first level, or we can promote two particles to the second lowest level. There is only one way to do this for either, which is to promote one particle from the lowest level to the second lowest level. There are ten ways to do the first arrangement (where the two particles are promoted to the second and third lowest levels above the last filled level) and five ways to do the second arrangement (where the last particle is promoted to the third level above the last filled level) for indistinguishable particles or fermions, as shown graphically:

Consider a particle promoted to the second lowest level with three energy units above the ground state. We can promote one particle up to three energy levels for distinguishable particles or bosons, leaving four particles in the first level, or promote one particle to the second lowest level and one to the third lowest level, or promote three particles to the second lowest level, but there is only one way to do this for each. There are ten ways to make the first arrangement, twenty ways to do the second arrangement, and five ways to do the third arrangement for indistinguishable particles or fermions, and these arrangements are depicted graphically as:

Part (d) Step 1: Given Information

We have to find the behavior of the bosonic system differ from that of the system of distinguishable particles.

Part (d) Step 2: Simplify

The Boltzmann factor is proportional to the likelihood that a system with a temperature of $T$, i.e.

$\mathcal{P}\propto e^{-E/kT}$

The energy of any system is the same, therefore this factor is the same, but when we factor in degeneracy (from the previous section, the degeneracy of the bosons is 3 and the degeneracy of the fermions is 35), we can see that at low temperatures, the fermions are more likely to be found than the bosons. The bosons are also likely to be discovered in the ground state.