Study of Thermodynamic Properties of $N$ Particles in $1$-Dimensional Harmonic Oscillator Potential

We have calculated the partition function and hence the energy and the specific heat of a two-particle system in $n$ numbers of finite nondegenerate energy levels of equal spacing distributed according to classical and quantum statistics. We have then extended our study for $N$ identical particles placed in $1$-D harmonic oscillator potential. It is observed that the specific heat of bosons and fermions are exactly the same in this potential at any temperature, and at high temperature the results are similar to that for the classical particles. The partition function of a system of two particles placed in $2$-D harmonic oscillator potential has also been determined as comparison.