# why for two level system we consider both energy level while finding number of bosons in ground state?

Let suppose we have N number of particles in two level system. .Effective number of cobosons in ground state is $$$$ that can be written as $$<\hat{n}_0>= Tr [\hat{n_0}\rho ]$$ where $$\hat{n}_0$$ is number operator of particle while $$\rho$$ is density operator. $$<\hat{n}_0>= \sum\limits_{n=0}^{N}$$ $$e^{-(E_0 )n/kT} e^{-(N-n)E_1/kT}/Z . \rho$$
where $$Z = \sum\limits_{n=0}^{N}$$ $$e^{-(E_0 )n/kT} e^{-(N-n)E_1/kT}$$

For multilevel system like 3D harmonic trap

$$<\hat{N}_m>= \sum\limits_{n=0}^{\infty}$$ $$e^{-(E_m -\mu)n/kT}/Z_m . \rho$$
where $$Z_m =(1- e^{-(E_m -\mu)/kT} )^{-1}$$. My question is that why for two level system they considered both energy level $$E_0$$ and $$E_1$$ but for multilevel they consider specific energy level $$E_m$$ in which we want to find effective number of particles.

• where did you find this from? I don't really understand what you're saying, so maybe looking at the original reference may help. – SuperCiocia Dec 13 '18 at 0:38