Question

16 , Eo Problem 1 (8 pts): An experimentalist is examining a kind of non-interacting identical particles that could be either spinless bosons or spin-half fermions by putting a number of them inside a potential and measuring the energy levels of the system, but without being able to resolve the degeneracy of each level. Energy levels do not depend on the particles' spin. The following values of the energy are observed: No particles: 0 -5€ 1 particle: E, 2E, 5E 2 particles: 2E, 3E, 4E, 6E, 7E, 10E 3 particles: 4E, 5E, 7E, 8E, 9E, 11E, 12E a. Deduce which kind of particles they are (bosons or fermions). b. Deduce the degeneracies of each of the observed energy values c. Write the wavefunction of the 4E three-particle state in terms of one-particle energy eigenfunctions. Include spin, if they have. (Use eigenstates u and d of the spin Sz.)

Answer 1

If a single particle is placed in the system. It can occupy any of the energy level and every level of energy can be observed. Therefore, The energy levels of single particle are shown as follows: If more than one particle occupy the system. The total energy of the particle shall be the combination of above-shown energy levels. Since, particles and identical or indistinguishable. Therefore, If 2 particles are placed in the system and 2E, 3E, 4E, 6E, 7E, and 10E energy levels are observed, then the following arrangement of particles is present for each energy.

2E 3E 4E 6E ZE 10E — Total energy Energy levels If 3 particles are placed in the system and 4E, 5E, 7E,SE, 9E, 11E, and 12E energy levels are observed, then the following arrangement of particles is present for each energy. 4E 5E ZE 8E 9E 11E 12E — Total energy 5E Energy levels 2E

As it is observed that no energy level is occupied by more than 2 particles. Since, for fermions, only two particles of opposite spin can occupy a single level. The particles must be fermions. Degeneracy of a level gives the number of arrangement of particles that can give the same energy level. It is observed that each level has unique arrangement of particles except 7E energy state for 3 particles. Therefore, each level has degeneracy equal to 1 except 7E energy state for 3 particles. 7E energy state for 3 particles has degeneracy level 2. The 4E energy state for 3 particles has 1 fermion in 2E energy state and 2 fermions in E energy state. Consider the single particle in 2E state has wavefunction u) or u). Since single particle is present and has no interaction with other particles. Its spin cannot be determined. For 2 particles present in E energy state. Both particles interact with each other on basis of spin. Consider one particle has state u). Therefore, other particle has stated). The combination of these u) and d) states, while considering their spin, gives their superposition state. Which is antisymmetric and given as follows: ø= Q(u,u,d)-(ud,u)+ (d,d,u) - (du,d)

All three particles cannot have same spin.